Cross product and applications: VCE Specialist Mathematics Unit 3

What this dot point is asking

VCAA wants you to compute the cross product of two three-dimensional vectors, understand that it produces a vector perpendicular to both, interpret its magnitude as an area, and apply it to find normal vectors, areas of parallelograms and triangles, and to test whether two vectors are parallel. The cross product underpins the equation of a plane studied alongside it.

Definition in components

For $\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} + a_3\mathbf{k}$ and $\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} + b_3\mathbf{k}$, the cross product is

A reliable way to compute it is the determinant expansion

expanding along the top row and remembering the middle term carries a minus sign.

Direction and magnitude

The vector $\mathbf{a} \times \mathbf{b}$ is perpendicular to both $\mathbf{a}$ and $\mathbf{b}$, so it is normal to the plane they span. Its direction follows the right-hand rule: point the fingers from $\mathbf{a}$ toward $\mathbf{b}$ and the thumb gives the direction of the product. Reversing the order reverses the direction, so $\mathbf{b} \times \mathbf{a} = -(\mathbf{a} \times \mathbf{b})$; the cross product is not commutative.

Its magnitude is

which equals the area of the parallelogram with sides $\mathbf{a}$ and $\mathbf{b}$.

Applications

Normal vector

Since $\mathbf{a} \times \mathbf{b}$ is perpendicular to both, it provides a normal to the plane containing them, which is exactly what is needed to write the plane's equation.

Area

The parallelogram on $\mathbf{a}$ and $\mathbf{b}$ has area $|\mathbf{a} \times \mathbf{b}|$, and the triangle with those two sides has area $\frac{1}{2}|\mathbf{a} \times \mathbf{b}|$.

Parallel test

If $\mathbf{a}$ and $\mathbf{b}$ are parallel, $\theta = 0$ or $\pi$, so $\sin\theta = 0$ and $\mathbf{a} \times \mathbf{b} = \mathbf{0}$. Conversely a zero cross product (for non-zero vectors) means they are parallel.

Examples in context

Example 1. $\mathbf{i} \times \mathbf{j} = \mathbf{k}$, illustrating the right-hand rule for the standard basis.

Example 2. If $\mathbf{a} \times \mathbf{b} = \mathbf{0}$ for non-zero $\mathbf{a}, \mathbf{b}$, the vectors are parallel, so one is a scalar multiple of the other.

Try this

Q1. Compute $(1, 0, 0) \times (0, 1, 0)$. [1 mark]

• Cue. $(0, 0, 1)$.

Q2. Find the area of the parallelogram on $(2, 0, 0)$ and $(0, 3, 0)$. [2 marks]

• Cue. Cross product $(0, 0, 6)$, magnitude $6$.

Q3. Show $(2, 4, 6)$ and $(1, 2, 3)$ are parallel using the cross product. [2 marks]

• Cue. Cross product is $\mathbf{0}$, so they are parallel.