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<root><page> <!DOCTYPE html><html> <head> <title>DLMF: 1.6 Vectors and Vector-Valued Functions</title> <meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> <script type="text/javascript"></script> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="keywords" lang="en" content="Einstein summation convention, Einstein summation convention for vectors, Gauss’s theorem for vector-valued functions, Green’s theorem, Green’s theorem for vector-valued functions, Levi-Civita symbol, Levi-Civita symbol for vectors, Stokes’ theorem, Stokes’ theorem for vector-valued functions, angle, area, boundary points, closed point set, cross product, curl, curve, del operator, divergence, divergence (or Gauss’s) theorem, divergence theorem, dot product, functions, gradient, integral over, integrals, integrals of vector-valued functions, length, line, line integral, magnitude, notations, of revolution, open point set, orientation, orientation-preserving, orientation-reversing, over parametrized surface, parallelepiped, parallelogram, parametrized surfaces, path, path integral, piecewise differentiable, piecewise differentiable curve, reparametrization of integration paths, right-hand rule, right-hand rule for cross products, scalar product, simple closed, simple closed curve, smooth, sphere, surface, tangent vector, three dimensions, two dimensions, unit, vector product, vector-valued, vector-valued functions, vectors, volume"> </head> <text><body class="color_default textfont_default titlefont_default fontsize_default navbar_default"> <div class="ltx_page_navbar"> <div class="ltx_page_navlogo"></div> <div class="ltx_page_navitems"> <form method="get" action="./search/search"> <ul> <li></li> <li></li> <li><small><input type="text" name="q" value="" size="6" class="ltx_page_navitem_search"><button type="submit">Search</button></small></li> <li></li> <li></li> <li></li> <li id="showinfo"></li> <li id="hideinfo"></li> </ul> </form> </div> <div class="ltx_page_navsponsors"> <div></div> </div> </div> <div class="ltx_page_main"> <div class="ltx_page_header"> </div> <div class="ltx_page_content"> <section class="ltx_section ltx_leqno"> <h1 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">§1.6 </span>Vectors and Vector-Valued Functions</h1> <div id="info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Keywords:</dt> <dd> </dd> <dt>Referenced by:</dt> <dd></dd> <dt>Permalink:</dt> <dd></dd> <dt>See also:</dt> <dd>Annotations for </dd> </dl> </div> </div> <h6>Contents</h6> <ul class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry"></li> <li class="ltx_tocentry"></li> <li class="ltx_tocentry"></li> <li class="ltx_tocentry"></li> <li class="ltx_tocentry"></li> </ul> <section id="i" class="ltx_subsection"> <h2 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">§1.6(i) </span>Vectors</h2> <div id="SS1.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Notes:</dt> <dd>See <cite class="ltx_cite ltx_citemacro_citet">Marsden and Tromba (, Chapter 1)</cite>. For (, pp. 82–84)</cite>.</dd> <dt>Permalink:</dt> <dd></dd> <dt>See also:</dt> <dd>Annotations for and </dd> </dl> </div> </div> <div id="SS1.p1" class="ltx_para"> <table id="E1" class="ltx_equationgroup ltx_eqn_table"> <tr id="Ex1" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="2" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.1</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m22.png" altimg-height="14px" altimg-valign="-2px" altimg-width="17px" alttext="\displaystyle\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m6.png" altimg-height="25px" altimg-valign="-7px" altimg-width="124px" alttext="\displaystyle=(a_{1},a_{2},a_{3})," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><msub><mi>a</mi><mn>1</mn></msub><mo>,</mo><msub><mi>a</mi><mn>2</mn></msub><mo>,</mo><msub><mi>a</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr id="Ex2" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m24.png" altimg-height="19px" altimg-valign="-2px" altimg-width="19px" alttext="\displaystyle\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m7.png" altimg-height="25px" altimg-valign="-7px" altimg-width="118px" alttext="\displaystyle=(b_{1},b_{2},b_{3})." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><msub><mi>b</mi><mn>1</mn></msub><mo>,</mo><msub><mi>b</mi><mn>2</mn></msub><mo>,</mo><msub><mi>b</mi><mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr><td class="ltx_align_right" colspan="5"> <div id="E1.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , and </dd> </dl> </div> </div> </td></tr> </table> </div> <section id="Px1" class="ltx_paragraph"> <h3 class="ltx_title ltx_title_paragraph">Dot Product (or Scalar Product)</h3> <div id="Px1.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Keywords:</dt> <dd></dd> <dt>See also:</dt> <dd>Annotations for , and </dd> </dl> </div> </div> <div id="Px1.p1" class="ltx_para"> <table id="E2" class="ltx_equation ltx_eqn_table"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.2</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E2.png" altimg-height="22px" altimg-valign="-5px" altimg-width="236px" alttext="\mathbf{a}\cdot\mathbf{b}=a_{1}b_{1}+a_{2}b_{2}+a_{3}b_{3}." display="block"><mrow><mrow><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow><mo>=</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>⁢</mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>⁢</mo><msub><mi>b</mi><mn>3</mn></msub></mrow></mrow></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr><td class="ltx_align_right" colspan="4"> <div id="E2.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , , and </dd> </dl> </div> </div> </td></tr> </table> </div> </section> <section id="Px2" class="ltx_paragraph"> <h3 class="ltx_title ltx_title_paragraph">Magnitude and Angle of Vector <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> </h3> <div id="Px2.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Keywords:</dt> <dd></dd> <dt>See also:</dt> <dd>Annotations for , and </dd> </dl> </div> </div> <div id="Px2.p1" class="ltx_para"> <table id="E3" class="ltx_equation ltx_eqn_table"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.3</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E3.png" altimg-height="27px" altimg-valign="-7px" altimg-width="122px" alttext="\|\mathbf{a}\|=\sqrt{\mathbf{a}\cdot\mathbf{a}}," display="block"><mrow><mrow><mrow><mo>∥</mo><mi mathvariant="bold">a</mi><mo>∥</mo></mrow><mo>=</mo><msqrt><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">a</mi></mrow></msqrt></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr><td class="ltx_align_right" colspan="4"> <div id="E3.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , , and </dd> </dl> </div> </div> </td></tr> </table> <table id="E4" class="ltx_equation ltx_eqn_table"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.4</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E4.png" altimg-height="51px" altimg-valign="-21px" altimg-width="152px" alttext="\cos\theta=\frac{\mathbf{a}\cdot\mathbf{b}}{\|\mathbf{a}\|\;\|\mathbf{b}\|};" display="block"><mrow><mrow><mrow><mi href="./4.14#E2">cos</mi><mo>⁡</mo><mi href="./1.6#Px2.p1">θ</mi></mrow><mo>=</mo><mfrac><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow><mrow><mrow><mo>∥</mo><mi mathvariant="bold">a</mi><mo rspace="5.3pt">∥</mo></mrow><mo>⁢</mo><mrow><mo>∥</mo><mi mathvariant="bold">b</mi><mo>∥</mo></mrow></mrow></mfrac></mrow><mo>;</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr><td class="ltx_align_right" colspan="4"> <div id="E4.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Symbols:</dt> <dd> <a href="./4.14#E2" title="(4.14.2) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m60.png" altimg-height="13px" altimg-valign="-2px" altimg-width="44px" alttext="\cos\NVar{z}" display="inline"><mrow><mi href="./4.14#E2">cos</mi><mo>⁡</mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: cosine function</a> and <a href="./1.6#Px2.p1" title="Magnitude and Angle of Vector a ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.6#Px2.p1">θ</mi></math>: angle between <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math></a> </dd> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , , and </dd> </dl> </div> </div> </td></tr> </table> <p class="ltx_p"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.6#Px2.p1">θ</mi></math> is the angle between <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math>.</p> </div> </section> <section id="Px3" class="ltx_paragraph"> <h3 class="ltx_title ltx_title_paragraph">Unit Vectors</h3> <div id="Px3.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Keywords:</dt> <dd></dd> <dt>See also:</dt> <dd>Annotations for , and </dd> </dl> </div> </div> <div id="Px3.p1" class="ltx_para"> <table id="E5" class="ltx_equationgroup ltx_eqn_table"> <tr id="Ex3" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.5</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m28.png" altimg-height="19px" altimg-valign="-2px" altimg-width="12px" alttext="\displaystyle\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(1,0,0)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr id="Ex4" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m31.png" altimg-height="23px" altimg-valign="-6px" altimg-width="13px" alttext="\displaystyle\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(0,1,0)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr id="Ex5" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m34.png" altimg-height="19px" altimg-valign="-2px" altimg-width="18px" alttext="\displaystyle\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m2.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(0,0,1)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr><td class="ltx_align_right" colspan="5"> <div id="E5.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Defines:</dt> <dd> <span class="ltx_text"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector (locally)</span>, <span class="ltx_text"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector (locally)</span> and <span class="ltx_text"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector (locally)</span> </dd> <dt>Referenced by:</dt> <dd></dd> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , , and </dd> </dl> </div> </div> </td></tr> </table> <table id="E6" class="ltx_equation ltx_eqn_table"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.6</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E6.png" altimg-height="23px" altimg-valign="-6px" altimg-width="182px" alttext="\mathbf{a}=a_{1}\mathbf{i}+a_{2}\mathbf{j}+a_{3}\mathbf{k}." display="block"><mrow><mrow><mi mathvariant="bold">a</mi><mo>=</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>⁢</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>⁢</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>⁢</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr><td class="ltx_align_right" colspan="4"> <div id="E6.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Symbols:</dt> <dd> <a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>, <a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and <a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a> </dd> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , , and </dd> </dl> </div> </div> </td></tr> </table> </div> </section> <section id="Px4" class="ltx_paragraph"> <h3 class="ltx_title ltx_title_paragraph">Cross Product (or Vector Product)</h3> <div id="Px4.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Keywords:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , and </dd> </dl> </div> </div> <div id="Px4.p1" class="ltx_para"> <table id="E7" class="ltx_equationgroup ltx_eqn_table"> <tr id="Ex6" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.7</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m29.png" altimg-height="23px" altimg-valign="-6px" altimg-width="44px" alttext="\displaystyle\mathbf{i}\times\mathbf{j}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">i</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m18.png" altimg-height="23px" altimg-valign="-6px" altimg-width="45px" alttext="\displaystyle=\mathbf{k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr id="Ex7" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m33.png" altimg-height="23px" altimg-valign="-6px" altimg-width="49px" alttext="\displaystyle\mathbf{j}\times\mathbf{k}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">j</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m16.png" altimg-height="23px" altimg-valign="-6px" altimg-width="39px" alttext="\displaystyle=\mathbf{i}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr id="Ex8" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m35.png" altimg-height="21px" altimg-valign="-4px" altimg-width="49px" alttext="\displaystyle\mathbf{k}\times\mathbf{i}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">k</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m17.png" altimg-height="23px" altimg-valign="-6px" altimg-width="40px" alttext="\displaystyle=\mathbf{j}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr><td class="ltx_align_right" colspan="5"> <div id="E7.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Symbols:</dt> <dd> <a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>, <a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and <a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a> </dd> <dt>Referenced by:</dt> <dd></dd> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , , and </dd> </dl> </div> </div> </td></tr> </table> <table id="E8" class="ltx_equationgroup ltx_eqn_table"> <tr id="Ex9" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.8</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m32.png" altimg-height="23px" altimg-valign="-6px" altimg-width="44px" alttext="\displaystyle\mathbf{j}\times\mathbf{i}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">j</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m10.png" altimg-height="23px" altimg-valign="-6px" altimg-width="60px" alttext="\displaystyle=-\mathbf{k}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr id="Ex10" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m36.png" altimg-height="23px" altimg-valign="-6px" altimg-width="49px" alttext="\displaystyle\mathbf{k}\times\mathbf{j}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">k</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m8.png" altimg-height="23px" altimg-valign="-6px" altimg-width="54px" alttext="\displaystyle=-\mathbf{i}," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr id="Ex11" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m30.png" altimg-height="21px" altimg-valign="-4px" altimg-width="49px" alttext="\displaystyle\mathbf{i}\times\mathbf{k}" display="inline"><mrow><mi href="./1.6#E5" mathvariant="bold">i</mi><mo>×</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m9.png" altimg-height="23px" altimg-valign="-6px" altimg-width="55px" alttext="\displaystyle=-\mathbf{j}." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo>-</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr><td class="ltx_align_right" colspan="5"> <div id="E8.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Symbols:</dt> <dd> <a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>, <a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and <a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a> </dd> <dt>Referenced by:</dt> <dd></dd> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , , and </dd> </dl> </div> </div> </td></tr> </table> <table id="E9" class="ltx_equation ltx_eqn_table"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.9</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_Math ltx_eqn_cell ltx_align_center"><math class="ltx_Math" alttext="\mathbf{a}\times\mathbf{b}=\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\ a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\end{vmatrix}\\ =(a_{2}b_{3}-a_{3}b_{2})\mathbf{i}+(a_{3}b_{1}-a_{1}b_{3})\mathbf{j}+(a_{1}b_{% 2}-a_{2}b_{1})\mathbf{k}\\ =\|\mathbf{a}\|\|\mathbf{b}\|(\sin\theta)\mathbf{n}," display="block"><mrow><mtable align="baseline 1" columnalign="left" columnspacing="0.1em" displaystyle="true"><mtr><mtd><mrow><mi mathvariant="bold">a</mi><mo>×</mo><mi mathvariant="bold">b</mi></mrow></mtd><mtd><mo>=</mo><mrow><mo href="./1.3#i">|</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">i</mi></mtd><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">j</mi></mtd><mtd columnalign="center"><mi href="./1.6#E5" mathvariant="bold">k</mi></mtd></mtr><mtr><mtd columnalign="center"><msub><mi>a</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>2</mn></msub></mtd><mtd columnalign="center"><msub><mi>a</mi><mn>3</mn></msub></mtd></mtr><mtr><mtd columnalign="center"><msub><mi>b</mi><mn>1</mn></msub></mtd><mtd columnalign="center"><msub><mi>b</mi><mn>2</mn></msub></mtd><mtd columnalign="center"><msub><mi>b</mi><mn>3</mn></msub></mtd></mtr></mtable><mo href="./1.3#i">|</mo></mrow><mo>=</mo><mrow><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>⁢</mo><msub><mi>b</mi><mn>3</mn></msub></mrow><mo>-</mo><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>⁢</mo><msub><mi>b</mi><mn>2</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>⁢</mo><mi href="./1.6#E5" mathvariant="bold">i</mi></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>3</mn></msub><mo>⁢</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo>-</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>b</mi><mn>3</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>⁢</mo><mi href="./1.6#E5" mathvariant="bold">j</mi></mrow><mo>+</mo><mrow><mrow><mo stretchy="false">(</mo><mrow><mrow><msub><mi>a</mi><mn>1</mn></msub><mo>⁢</mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo>-</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mo>⁢</mo><msub><mi>b</mi><mn>1</mn></msub></mrow></mrow><mo stretchy="false">)</mo></mrow><mo>⁢</mo><mi href="./1.6#E5" mathvariant="bold">k</mi></mrow></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mo>=</mo><mrow><mrow><mo>∥</mo><mi mathvariant="bold">a</mi><mo>∥</mo></mrow><mo>⁢</mo><mrow><mo>∥</mo><mi mathvariant="bold">b</mi><mo>∥</mo></mrow><mo>⁢</mo><mrow><mo stretchy="false">(</mo><mrow><mi href="./4.14#E1">sin</mi><mo>⁡</mo><mi href="./1.6#Px2.p1">θ</mi></mrow><mo stretchy="false">)</mo></mrow><mo>⁢</mo><mi mathvariant="bold">n</mi></mrow><mo>,</mo></mtd></mtr></mtable></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr><td class="ltx_align_right" colspan="4"> <div id="E9.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Symbols:</dt> <dd> <a href="./1.3#i" title="§1.3(i) Definitions and Elementary Properties ‣ §1.3 Determinants ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m63.png" altimg-height="18px" altimg-valign="-2px" altimg-width="32px" alttext="\det" display="inline"><mo href="./1.3#i">det</mo></math>: determinant</a>, <a href="./4.14#E1" title="(4.14.1) ‣ §4.14 Definitions and Periodicity ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions" class="ltx_ref"><math class="ltx_Math" altimg="m103.png" altimg-height="18px" altimg-valign="-2px" altimg-width="42px" alttext="\sin\NVar{z}" display="inline"><mrow><mi href="./4.14#E1">sin</mi><mo>⁡</mo><mi class="ltx_nvar" href="./4.1#p2.t1.r4">z</mi></mrow></math>: sine function</a>, <a href="./1.6#Px2.p1" title="Magnitude and Angle of Vector a ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m104.png" altimg-height="18px" altimg-valign="-2px" altimg-width="14px" alttext="\theta" display="inline"><mi href="./1.6#Px2.p1">θ</mi></math>: angle between <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math></a>, <a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m85.png" altimg-height="18px" altimg-valign="-2px" altimg-width="11px" alttext="\mathbf{i}" display="inline"><mi href="./1.6#E5" mathvariant="bold">i</mi></math>: unit vector</a>, <a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m86.png" altimg-height="21px" altimg-valign="-6px" altimg-width="11px" alttext="\mathbf{j}" display="inline"><mi href="./1.6#E5" mathvariant="bold">j</mi></math>: unit vector</a> and <a href="./1.6#E5" title="(1.6.5) ‣ Unit Vectors ‣ §1.6(i) Vectors ‣ §1.6 Vectors and Vector-Valued Functions ‣ Areas ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m87.png" altimg-height="18px" altimg-valign="-2px" altimg-width="16px" alttext="\mathbf{k}" display="inline"><mi href="./1.6#E5" mathvariant="bold">k</mi></math>: unit vector</a> </dd> <dt>Referenced by:</dt> <dd></dd> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , , and </dd> </dl> </div> </div> </td></tr> </table> <p class="ltx_p">where <math class="ltx_Math" altimg="m88.png" altimg-height="13px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{n}" display="inline"><mi mathvariant="bold">n</mi></math> is the unit vector normal to <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math> whose direction is determined by the right-hand rule; see Figure .</p> </div> <figure id="F1" class="ltx_figure"><figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure">Figure 1.6.1: </span>Vector notation. Right-hand rule for cross products. </figcaption> <div id="F1.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Referenced by:</dt> <dd></dd> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , , and </dd> </dl> </div> </div> </figure> <div id="Px4.p2" class="ltx_para"> <p class="ltx_p">Area of parallelogram with vectors <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math> and <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math> as sides <math class="ltx_Math" altimg="m43.png" altimg-height="23px" altimg-valign="-7px" altimg-width="93px" alttext="=\|\mathbf{a}\times\mathbf{b}\|" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>∥</mo><mrow><mi mathvariant="bold">a</mi><mo>×</mo><mi mathvariant="bold">b</mi></mrow><mo>∥</mo></mrow></mrow></math>. </p> </div> <div id="Px4.p3" class="ltx_para"> <p class="ltx_p">Volume of a parallelepiped with vectors <math class="ltx_Math" altimg="m75.png" altimg-height="13px" altimg-valign="-2px" altimg-width="15px" alttext="\mathbf{a}" display="inline"><mi mathvariant="bold">a</mi></math>, <math class="ltx_Math" altimg="m77.png" altimg-height="18px" altimg-valign="-2px" altimg-width="17px" alttext="\mathbf{b}" display="inline"><mi mathvariant="bold">b</mi></math>, and <math class="ltx_Math" altimg="m81.png" altimg-height="13px" altimg-valign="-2px" altimg-width="14px" alttext="\mathbf{c}" display="inline"><mi mathvariant="bold">c</mi></math> as edges <math class="ltx_Math" altimg="m42.png" altimg-height="23px" altimg-valign="-7px" altimg-width="125px" alttext="=\left|\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})\right|" display="inline"><mrow><mi></mi><mo>=</mo><mrow><mo>|</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">b</mi><mo>×</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>|</mo></mrow></mrow></math>. </p> <table id="EGx1" class="ltx_equationgroup ltx_eqn_table"> <tbody id="E10"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.10</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m23.png" altimg-height="25px" altimg-valign="-7px" altimg-width="104px" alttext="\displaystyle\mathbf{a}\times(\mathbf{b}\times\mathbf{c})" display="inline"><mrow><mi mathvariant="bold">a</mi><mo>×</mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">b</mi><mo>×</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m15.png" altimg-height="25px" altimg-valign="-7px" altimg-width="185px" alttext="\displaystyle=\mathbf{b}(\mathbf{a}\cdot\mathbf{c})-\mathbf{c}(\mathbf{a}\cdot% \mathbf{b})," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi mathvariant="bold">b</mi><mo>⁢</mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi mathvariant="bold">c</mi><mo>⁢</mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr><td class="ltx_align_right" colspan="5"> <div id="E10.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , , and </dd> </dl> </div> </div> </td></tr> </tbody> <tbody id="E11"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.11</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m1.png" altimg-height="25px" altimg-valign="-7px" altimg-width="104px" alttext="\displaystyle(\mathbf{a}\times\mathbf{b})\times\mathbf{c}" display="inline"><mrow><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">a</mi><mo>×</mo><mi mathvariant="bold">b</mi></mrow><mo stretchy="false">)</mo></mrow><mo>×</mo><mi mathvariant="bold">c</mi></mrow></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m14.png" altimg-height="25px" altimg-valign="-7px" altimg-width="185px" alttext="\displaystyle=\mathbf{b}(\mathbf{a}\cdot\mathbf{c})-\mathbf{a}(\mathbf{b}\cdot% \mathbf{c})." display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mrow><mi mathvariant="bold">b</mi><mo>⁢</mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow><mo>-</mo><mrow><mi mathvariant="bold">a</mi><mo>⁢</mo><mrow><mo stretchy="false">(</mo><mrow><mi mathvariant="bold">b</mi><mo>⋅</mo><mi mathvariant="bold">c</mi></mrow><mo stretchy="false">)</mo></mrow></mrow></mrow></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr><td class="ltx_align_right" colspan="5"> <div id="E11.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , , and </dd> </dl> </div> </div> </td></tr> </tbody> </table> </div> </section> </section> <section id="ii" class="ltx_subsection"> <h2 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">§1.6(ii) </span>Vectors: Alternative Notations</h2> <div id="SS2.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Keywords:</dt> <dd></dd> <dt>Permalink:</dt> <dd></dd> <dt>See also:</dt> <dd>Annotations for and </dd> </dl> </div> </div> <div id="SS2.p1" class="ltx_para"> <p class="ltx_p">The following notations are often used in the physics literature; see for example <cite class="ltx_cite ltx_citemacro_citet">Lorentz<span class="ltx_text ltx_bib_etal"> et al.</span> (, pp. 122–123)</cite>.</p> </div> <section id="Px5" class="ltx_paragraph"> <h3 class="ltx_title ltx_title_paragraph">Einstein Summation Convention</h3> <div id="Px5.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Keywords:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , and </dd> </dl> </div> </div> <div id="Px5.p1" class="ltx_para"> <p class="ltx_p">Much vector algebra involves summation over suffices of products of vector components. In almost all cases of repeated suffices, we can suppress the summation notation entirely, if it is understood that an implicit sum is to be taken over any repeated suffix. Thus pairs of indefinite suffices in an expression are resolved by being summed over (or “traced” over).</p> </div> </section> <section id="Px6" class="ltx_paragraph"> <h3 class="ltx_title ltx_title_paragraph">Example</h3> <div id="Px6.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>See also:</dt> <dd>Annotations for , and </dd> </dl> </div> </div> <div id="Px6.p1" class="ltx_para"> <table id="E12" class="ltx_equation ltx_eqn_table"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.12</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E12.png" altimg-height="70px" altimg-valign="-30px" altimg-width="207px" alttext="a_{j}b_{j}=\sum_{j=1}^{3}a_{j}b_{j}=\mathbf{a}\cdot\mathbf{b}." display="block"><mrow><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo>⁢</mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow><mo>=</mo><mrow><munderover><mo largeop="true" movablelimits="false" symmetric="true">∑</mo><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo>=</mo><mn>1</mn></mrow><mn>3</mn></munderover><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo>⁢</mo><msub><mi>b</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow></mrow><mo>=</mo><mrow><mi mathvariant="bold">a</mi><mo>⋅</mo><mi mathvariant="bold">b</mi></mrow></mrow><mo>.</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr><td class="ltx_align_right" colspan="4"> <div id="E12.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Symbols:</dt> <dd><a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a></dd> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , , and </dd> </dl> </div> </div> </td></tr> </table> </div> <div id="Px6.p2" class="ltx_para"> <p class="ltx_p">Next,</p> <table id="E13" class="ltx_equationgroup ltx_eqn_table"> <tr id="Ex12" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="3" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equationgroup ltx_align_left">1.6.13</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m25.png" altimg-height="17px" altimg-valign="-5px" altimg-width="25px" alttext="\displaystyle\mathbf{e}_{1}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mn>1</mn></msub></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m5.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(1,0,0)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr id="Ex13" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m26.png" altimg-height="17px" altimg-valign="-5px" altimg-width="25px" alttext="\displaystyle\mathbf{e}_{2}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mn>2</mn></msub></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m4.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(0,1,0)," display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></mrow><mo>,</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr id="Ex14" class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math class="ltx_Math" altimg="m27.png" altimg-height="17px" altimg-valign="-5px" altimg-width="25px" alttext="\displaystyle\mathbf{e}_{3}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mn>3</mn></msub></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math class="ltx_Math" altimg="m3.png" altimg-height="25px" altimg-valign="-7px" altimg-width="96px" alttext="\displaystyle=(0,0,1);" display="inline"><mrow><mrow><mi></mi><mo>=</mo><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow><mo>;</mo></mrow></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr><td class="ltx_align_right" colspan="5"> <div id="E13.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Defines:</dt> <dd><span class="ltx_text"><math class="ltx_Math" altimg="m84.png" altimg-height="18px" altimg-valign="-8px" altimg-width="23px" alttext="\mathbf{e}_{j}" display="inline"><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></math>: unit vectors (locally)</span></dd> <dt>Symbols:</dt> <dd><a href="./1.1#p2.t1.r4" title="§1.1 Special Notation ‣ Notation ‣ Chapter 1 Algebraic and Analytic Methods" class="ltx_ref"><math class="ltx_Math" altimg="m117.png" altimg-height="20px" altimg-valign="-6px" altimg-width="14px" alttext="j" display="inline"><mi href="./1.1#p2.t1.r4">j</mi></math>: integer</a></dd> <dt>Permalink:</dt> <dd></dd> <dt>Encodings:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , , and </dd> </dl> </div> </div> </td></tr> </table> <p class="ltx_p">compare (). Thus <math class="ltx_Math" altimg="m108.png" altimg-height="18px" altimg-valign="-8px" altimg-width="80px" alttext="a_{j}\mathbf{e}_{j}=\mathbf{a}" display="inline"><mrow><mrow><msub><mi>a</mi><mi href="./1.1#p2.t1.r4">j</mi></msub><mo>⁢</mo><msub><mi href="./1.6#E13" mathvariant="bold">e</mi><mi href="./1.1#p2.t1.r4">j</mi></msub></mrow><mo>=</mo><mi mathvariant="bold">a</mi></mrow></math>.</p> </div> </section> <section id="Px7" class="ltx_paragraph"> <h3 class="ltx_title ltx_title_paragraph">Levi-Civita Symbol</h3> <div id="Px7.info" class="ltx_metadata ltx_info"> <div class="ltx_infocontent"> <dl> <dt>Keywords:</dt> <dd> </dd> <dt>See also:</dt> <dd>Annotations for , and </dd> </dl> </div> </div> <div id="Px7.p1" class="ltx_para"> <table id="E14" class="ltx_equation ltx_eqn_table"> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_left"><span class="ltx_tag ltx_tag_equation ltx_align_left">1.6.14</span></td> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math class="ltx_Math" altimg="E14.png" altimg-height="92px" altimg-valign="-40px" altimg-width="461px" alttext="\epsilon_{jk\ell}=\begin{cases}+1,&\text{if }j,k,\ell\text{ is even % permutation of }1,2,3,\\ -1,&\text{if }j,k,\ell\text{ is odd permutation of }1,2,3,\\ \phantom{-}0,&\text{otherwise}.\end{cases}" display="block"><mrow><msub><mi href="./1.6#E14">ϵ</mi><mrow><mi href="./1.1#p2.t1.r4">j</mi><mo href="./1.6#E14">⁣</mo><mi href="./1.1#p2.t1.r4">k</mi><mo href="./1.6#E14">⁣</mo><mi mathvariant="normal">ℓ</mi></mrow></msub><mo>=</mo><mrow><mo>{</mo><mtable columnspacing="5pt" displaystyle="true" rowspacing="0pt"><mtr><mtd columnalign="left"><mrow><mrow><mo>+</mo><mn>1</mn></mrow><mo>,</mo></mrow></mtd><mtd columnalign="left"><mrow><mrow><mrow><mtext>if </mtext><mo>⁢</mo><mi href="./1.1#p2.t1.r4">j</mi><