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11.8: Cross Product and Torque

  • Page ID
    88003
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

      By Carey Smith

      Cross product calculations are inherently 3-dimensional. The cross product of 2 vectors, a and b, is another vector, c, which is perpendicular to both a and b.

      When a and b are parallel, c is zero.

      When a and b are perpendicular, the magnitude of c = the product of the magnitudes of a and b.

      Watch this video!

      It gives a visual explanation of the cross product, including the right-hand rule. It also demonstrates the dot product.

      .

      Examples for parallel vectors, perpendicular vectors, and vectors at a 60 degree angle are given here. 3-D Plots are given for each example below, except for the zero case. These 2D plots use the plot3() function.

      Example \(\PageIndex{1}\) cross(a,b) a & b parallel

      The cross product of parallel vectors is zero.

      a = [1 2 3]
      b = 2*a % = [2 4 6]
      ab_cross = cross(a,b) % = [0 0 0]

      Solution

      Add example text here.

      .

      Example \(\PageIndex{2}\) cross(c1,d1) c1 & d1 perpendicular unit vectors

      c1 = [1 0 0]
      d1 = [0 1 0]
      cd1_cross = cross(c1,d1)
      % = [0 0 1]
      % Plot each vector
      figure;
      plot3([0,c1(1)],[0,c1(2)],[0,c1(3)],'-*')
      hold on;
      plot3([0,d1(1)],[0,d1(2)],[0,d1(3)],'-o')
      % Plot the cross-product vector on the same figure
      plot3([0,cd1_cross(1)],[0,cd1_cross(2)],[0,cd1_cross(3)],'k-d')
      axis equal
      grid on;
      xlabel('x')
      ylabel('y')
      zlabel('z')
      % Use tools->Rotate 3D to verify that cd1_cross is orthogonal
      % to the 1st 2 vectors and follows the right-hand rule.
      title('cd1\_cross = cross(c1,d1), perpendicular')

      Solution

      cross_c1_d1.png

      Add example text here.

      .

      Example \(\PageIndex{3}\) cross(c2,d2) c2 & d2 perpendicular, 45 degs

      c2 = [ 1 1 0] % at 45 degs
      d2 = [ 1 -1 0] % at -45 degs
      cd2_cross = cross(c2,d2) % = [0 0 -2]
      % Plot each vector
      figure;
      plot3([0,c2(1)],[0,c2(2)],[0,c2(3)],'-*')
      hold on;
      plot3([0,d2(1)],[0,d2(2)],[0,d2(3)],'-o')
      % Plot the cross-product vector on the same figure
      plot3([0,cd2_cross(1)],[0,cd2_cross(2)],[0,cd2_cross(3)],'k-d')
      axis equal
      grid on;
      xlabel('x')
      ylabel('y')
      zlabel('z')
      % Use tools->Rotate 3D to verify that cd2_cross is orthogonal
      % to the 1st 2 vectors and follows the right-hand rule.
      title('cd2\_cross = cross(c2,d2), perpendicular')

      Solution

      cross_c2_d2.png

      Add example text here.

      .

      Example \(\PageIndex{4}\) cross(e,f) angle between e and f = 60 degrees

      e1 = [1 0 0] % at 0 degs
      f1 = [cosd(60) sind(60) 0] % at 60 degs
      % f = [0.500 0.866 0]
      ef1_cross = cross(e1,f1)
      % = [0 0 0.866]
      % Plot each vector
      figure;
      plot3([0,e1(1)],[0,e1(2)],[0,e1(3)],'-*')
      hold on;
      plot3([0,f1(1)],[0,f1(2)],[0,f1(3)],'-o')
      % Plot the cross-product vector on the same figure
      plot3([0,ef1_cross(1)],[0,ef1_cross(2)],[0,ef1_cross(3)],'k-d')
      axis equal
      grid on;
      xlabel('x')
      ylabel('y')
      zlabel('z')
      % Use tools->Rotate 3D to verify that ef1_cross is orthogonal
      % to the 1st 2 vectors and follows the right-hand rule.
      title('ef2\_cross = cross(e1,f1), 60 degrees')

      Solution

      cross_e1_f1.png

      Add example text here.

      .

      Torque

      For a lever of length R, and a Force, F, Torque = RxF (R cross F).

      When the Force is in pounds and the length in feet, then the units are pound-feet.

      When the Force is in Newtons and the length in meters, then the units are Newton-meters.

      In mechanical engineering, torque is known as "moment".

      This video illustrates torque nicely: (Cross Product for Calculating Torque, http://BigBangPhysics.com)

      Exercise \(\PageIndex{1}\) Torque HW

      Consider a wrench loosening a bolt..
      Our coordinate system is:

      • x is horizontal in the plane of the wrench
      • y is horizontal perpendicular to x (into the paper)
      • z is vertical

      [2 pts] Set up the force and wrench vectors:

      A force is applied near the end of the wrench at a distance R = 0.20m from the bolt on the left side at a 20 degree angle from horizontal. When the bolt turns, it will have a counter-clockwise rotation.
      R = 0.20*[-cosd(20), 0, sind(20)] %(m)
      The magnitude of the force is 100N. The direction of the force is down.
      F = 100*[0, 0, -1] %(N)

      [4 pts] Compute the torque using the cross() function.
      torque = cross(R,F)

      [4 pts] Plot the R, F, and torque vectors on 1 figure using code similar to the cross product examples.
      Since the units of R, F, and torque are all different, do not use "axis equal".
      Add title('Torque')
      Use tools->Rotate 3D to verify that the cross product is orthogonal to the R and F vectors and follows the right-hand rule

      Answer

      Add texts here. Do not delete this text first.

      .


      This page titled 11.8: Cross Product and Torque is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Carey Smith.

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