# MA 227 Standard S03

The Cross Product

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)