The Vector Product (DP IB Analysis & Approaches (AA): HL): Revision Note

The vector (cross) product

What is the vector product?

• The vector product is an operation which takes two vectors and outputs a vector

• The vector product is perpendicular to both vectors

• The vector product between two vectors $\mathit{v}$ and $\mathit{w}$ is denoted $\mathit{v}\times \mathit{w}$

  • This is why it is also called the cross product

• The direction of the vector product follows the right-hand rule

  • Using you right hand:

    • Point your index finger in the direction of the first vector

    • Point your middle finger in the direction of the second vector

    • The direction of the vector product is given by the direction of your thumb

How is the vector product calculated?

• One formula for the vector product is $\mathit{v}\times \mathit{w}= \left(\begin{array}{c}{v}_2{w}_3- v_3{w}_2\\ v_3{w}_1- v_1{w}_3\\ v_1{w}_2- v_2{w}_1\end{array}\right)$ 

  • $\mathit{v}= \left(\genfrac{}{}{0pt}{}{v_1}{\begin{array}{c}{v}_2\\ v_3\end{array}}\right)$

  • $\mathit{w}= \left(\genfrac{}{}{0pt}{}{w_1}{\begin{array}{c}{w}_2\\ w_3\end{array}}\right)$

• Another formula for the vector product is $\mathit{v}\times \mathit{w}=\left|\mathit{v}\right|\left|\mathit{w}\right|\sin \theta \mathit{n}$

  • $\theta$ is the angle between $\mathit{v}$ and $\mathit{w}$

  • $\mathit{n}$ is a unit vector that is normal to the two vectors and follows the right-hand rule

What properties of the vector product do I need to know?

• The vector product is not commutative

  • $\mathit{v}\times \mathit{w}\ne \mathit{w}\times \mathit{v}$

• Changing the order of the vectors reverses the direction of the vector product

  • $\mathit{v}\times \mathit{w}=-\mathit{w}\times \mathit{v}$

• The distributive law over addition can be used to expand brackets

  •  $\mathit{u}\times \left(\mathit{v}+\mathit{w}\right)=\mathit{u}\times \mathit{v}+\mathit{u}\times \mathit{w}$

• The vector product is associative with respect to multiplication by a scalar

  • $\left(k\mathit{v}\right)\times \mathit{w}=\mathit{v}\times \left(k\mathit{w}\right)=k(\mathit{v}\times \mathit{w})$

• The vector product between a vector and itself is equal to the zero vector

  • $\mathit{v}\times \mathit{v}=\mathbf{0}$

• The vector product of two parallel vectors is equal to the zero vector

  • This is because $\sin 0°=\sin 180°=0$

• The converse is also true

  • If $\mathit{v}\times \mathit{w}=\mathbf{0}$ for non-zero vectors

  • Then $\mathit{v}$ and $\mathit{w}$ must be parallel

• The absolute value of the vector product of two perpendicular vectors is equal to the product of their magnitudes

  • $|\mathit{v}\times \mathit{w}|=\left|\mathit{v}\right|\left|\mathit{w}\right|$

    • This is because $\sin 90°=1$