Section 2.4 - The Cross Product

Section 2.4 - The Cross Product

Definition and Formula of the Cross Product

The cross product (also known as the vector product) is an operation on two vectors in three-dimensional space, resulting in a third vector that is perpendicular to both original vectors. If $\vec{}a=\left(a_1,a_2,a_3\right)$ and $\vec{}b=\left(b_1,b_2,b_3\right)$, then the cross product $\vec{}a\times\vec{}b$ is defined as:

• Formula:

$\vec{a} \times \vec{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$

$= (a_2b_3 - a_3b_2)\mathbf{i} - (a_1b_3 - a_3b_1)\mathbf{j} + (a_1b_2 - a_2b_1)\mathbf{k}$

$= (a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1)$

Examples of the Cross Product

• Example 1: If $\vec{}a=\left(1,3,4\right)$ and $\vec{}b=\left(2,7,-5\right)$, find $\vec{}a\times\vec{}b$.

• Example 2: Show that $\vec{a} \times \vec{a} = \vec{0}$ for any vector $\vec{}a$ in $\mathbb{R}^3$.

• Example 3: Find the area of the parallelogram with vertices $A\left(-3,0\right)$,$B\left(-1,3\right)$,$C\left(5,2\right)$, and$D\left(3,-1\right)$.

Geometric Interpretation and Right-Hand Rule

The direction of $\vec{}a\times\vec{}b$ is determined by the right-hand rule. If you point the fingers of your right hand in the direction of $\vec{}a$ and curl them toward $\vec{}b$, your thumb points in the direction of the cross product.

• Magnitude: The magnitude of the cross product is given by:

$|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin \theta$

where$\theta$is the angle between $\vec{}a$and $\vec{}b$.

• Application: The area of a parallelogram formed by vectors $\vec{}a$ and $\vec{}b$ is $\vec{}a\times\vec{}b$.

Properties of the Cross Product

• Anticommutativity: $\vec{}a\times\vec{}b=-\left(\vec{}b\times\vec{}a\right)$

• Distributivity: $\vec{}a\times\left(\vec{}b+\vec{}c\right)=\vec{}a\times\vec{}b+\vec{}a\times\vec{}c$

• Scalar Multiplication: If$c$ is a scalar, then$\left(c\vec{}a\right)\times\vec{}b=c\left(\vec{}a\times\vec{}b\right)$

• Zero Vector: $\vec{a} \times \vec{a} = \vec{0}$

• Orthogonality: The cross product $\vec{}a\times\vec{}b$ is orthogonal to both $\vec{}a$ and $\vec{}b$.

Summary Table: Properties of the Cross Product

Property	Equation	Description
Anticommutativity	$\vec{a} \times \vec{b} = - (\vec{b} \times \vec{a})$	Switching the order reverses the direction
Distributivity	$\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$	Cross product distributes over addition
Scalar Multiplication	$(c\vec{a}) \times \vec{b} = c(\vec{a} \times \vec{b})$	Scalar can be factored out
Zero Vector	$\vec{a} \times \vec{a} = \vec{0}$	Cross product of a vector with itself is zero
Orthogonality	$\vec{a} \cdot (\vec{a} \times \vec{b}) = 0$	Result is perpendicular to both vectors

Additional info:

• The cross product is only defined in three dimensions.

• It is widely used in physics and engineering, especially in calculations involving torque, angular momentum, and determining normal vectors to surfaces.