\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

MA 227 Standard S03

The Cross Product
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At the end of the course, each student should be able to…

  • S03: CrossProd. Compute and apply the cross product of two vectors.

S03. The Cross Product

  • The right-hand rule (RHR) is a method of consistently producing an orthogonal direction two a pair of vectors \(\vect v,\vect w\).
    • If \(\vect v\) is a right-hand thumb, and \(\vect w\) is a right-hand index finger, then the RHR gives the direction of the right-hand middle finger when extended orthogonally to the thumb and index.
  • The cross product \(\vect v\times\vect w\) measures the torque caused by a force given by \(\vect v\) on an arm given by \(\vect w\).
    • If the torque is orthogonal to the arm, then torque is simply given by the vector in the direction of the RHR (in the direction of “righty-tighty, lefty-loosy”) with a magitnude given by scalar multiplication of the magnitudes of \(\vect v,\vect w\).
    • If the force is parallel to the arm, then no torque is caused.
    • It may be computed by \(\vect v\times\vect w=(\|\vect v\|\|\vect w\|\sin\theta)\vect n\) where \(\theta\) is the angle between the vectors and \(\vect n\) is the unit vector direction given by the RHR.
  • The following properties may be proven about the cross product.
    • \(\vect{v} \times \vect{w} = -(\vect{w} \times \vect{v})\)
    • \((c\vect{v})\times \vect{w} = \vect{v} \times (c\vect{w}) = c(\vect{v} \times \vect{w})\)
    • \(\vect{v} \times (\vect{w}_1 + \vect{w}_2) = \vect{v} \times \vect{w}_1 + \vect{v} \times \vect{w}_2\)
    • \(\vect{v} \times \vect{v} = \vect{0}\)
    • \(\vect{0} \times \vect{v} = \vect{0}\)
    • \(\veci\times\vecj=\veck\), \(\vecj\times\veck=\veci\), and \(\veck\times\veci=\vecj\)
  • These properties may be used to obtain a simplified formula: \(\vect{v} \times \vect{w} = \<v_yw_z-v_zw_y,v_zw_x-v_xw_z,v_xw_y-v_yw_x\>\).
    • This in turn may be simplified as a determinant: \(\vect v\times\vect w = \detThree{\veci}{\vecj}{\veck}{v_x}{v_y}{v_z}{w_x}{w_y}{w_z}\).
    • Note \(\theta\) may be computed from \(\sin\theta=\frac{\|\vect v\times\vect w\|}{\|\vect v\|\|\vect w\|}\).
    • Thus two nonzero vectors are parallel if and only if \(\vect v\times\vect w=\vect 0\).

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 11.4 (exercises 1-14, 23-28)