Vector Product - Cross Product (2024)

(Vector Product of Two Vectors)

The cross product, also called vector product of two vectors is written \(\vec{u}\times \vec{v}\) and is the second way to multiply two vectors together.

When we multiply two vectors using the cross product we obtain a new vector. This is unlike the scalar product (or dot product) of two vectors, for which the outcome is a scalar (a number, not a vector!).

In fact, the cross product of two vectors \(\vec{u}\) and \(\vec{v}\) is a "new" vector that is perpendicular to both \(\vec{u}\) and \(\vec{v}\), we say that \(\vec{u}\times \vec{v}\) is normal to the plane containing \(\vec{u}\) and \(\vec{v}\).

How to Calculate the Cross Product

To calculate the vector product, or cross product, of two vectors we use either one of the following two options:

• Option 1: use the Formula (learn it off by heart)
• Option 2: use Matrix Algebra (recommended method)

We look at both options here.

Option 1 - The Formula: Vector Product \(\vec{u}\times \vec{v}\)

Given two vectors \(\vec{u} = \begin{pmatrix}u_1 \\ u_2 \\ u_3 \end{pmatrix}\) and \(\vec{v} = \begin{pmatrix}v_1 \\ v_2 \\ v_3 \end{pmatrix}\), the vector product, or cross product, \(\vec{u}\times \vec{v}\) can be calculated using the following formula:\[\vec{u} \times \vec{v} = \begin{pmatrix}u_2v_3 - v_2u_3\end{pmatrix} \vec{i} - \begin{pmatrix}u_1v_3 -v_1u_3\end{pmatrix} \vec{j} + \begin{pmatrix}u_1v_2 - v_1u_2\end{pmatrix} \vec{k}\]To avoid having the subtraction between the first and second term, some math courses (such as IB AA HL Mathematics) rewrite this formula as:\[\vec{u} \times \vec{v} = \begin{pmatrix}u_2v_3 - v_2u_3\end{pmatrix} \vec{i} + \begin{pmatrix} v_1u_3 - u_1v_3\end{pmatrix} \vec{j} + \begin{pmatrix}u_1v_2 - v_1u_2\end{pmatrix} \vec{k}\]Both formula are completely equivalent.

Example

Using the formula, stated above, find the cross product, \(\vec{u}\times \vec{v}\) of \(\vec{u} = \begin{pmatrix} 1 \\ -3 \\ 2 \end{pmatrix}\) and \(\vec{v} = \begin{pmatrix} 4 \\ 0 \\ 6 \end{pmatrix}\).

Solution

Using the second formula, stated above:\[\vec{u} \times \vec{v} = \begin{pmatrix}u_2v_3 - v_2u_3\end{pmatrix} \vec{i} + \begin{pmatrix} v_1u_3 - u_1v_3\end{pmatrix} \vec{j} + \begin{pmatrix}u_1v_2 - v_1u_2\end{pmatrix} \vec{k}\]With \(u_1 = 1\), \(u_2 = -3\), \(u_3 = 2\), \(v_1 = 4\), \(v_2 = 0\) and \(v_3 = 6\), we find:\[\begin{aligned}\vec{u}\times \vec{v} & =\begin{pmatrix}-3 \times 6 - 0 \times 2 \end{pmatrix}\vec{i} + \begin{pmatrix}4 \times 2 - 1 \times 6 \end{pmatrix}\vec{j} + \begin{pmatrix}1 \times 0 - 4 \times (-3)\end{pmatrix} \vec{k} \\& = \begin{pmatrix}-18 -0\end{pmatrix}\vec{i} + \begin{pmatrix}8 - 6 \end{pmatrix} \vec{j} + \begin{pmatrix}0+12\end{pmatrix}\vec{k} \\\vec{u}\times \vec{v} & = -18 \vec{i} + 2 \vec{j} + 12 \vec{k}\end{aligned} \]

Option 2: Matrix Algebra - Determinant of a 3 by 3 Matrix (recommended)

Given two vectors \(\vec{u} = \begin{pmatrix}u_1 \\ u_2 \\ u_3 \end{pmatrix}\) and \(\vec{v} = \begin{pmatrix}v_1 \\ v_2 \\ v_3 \end{pmatrix}\), the vector product, or cross product, \(\vec{u}\times \vec{v}\) equals to the following:\[\vec{u}\times \vec{v} = \begin{vmatrix}\vec{i} & \vec{j} & \vec{k} \\u_1 & u_2 & u_3 \\v_1 & v_2 & v_3\end{vmatrix}\]Notice that:

• the components in the first row consist of the unit base vectors \(\vec{i}\), \(\vec{j}\) and \(\vec{k}\) (this will always be the first row),
• the second row is the first vector in the product \(\vec{u}\times \vec{v}\), so in this case \(\vec{u}\),
• the third row is the second vector in the product \(\vec{u}\times \vec{v}\), so in this case \(\vec{v}\).

Tutorial: Cross Product - Determinant Method

Example

Given \(\vec{u} = \begin{pmatrix} 2 \\ 1 \\ -3 \end{pmatrix}\) and \(\vec{v} = \begin{pmatrix} 4 \\ 0 \\ 5 \end{pmatrix}\), calculate their cross product \(\vec{u}\times \vec{v}\).

Solution

Using the method we just saw, we can state:\[\begin{aligned}\vec{u} \times \vec{v} & = \begin{vmatrix}\vec{i} & \vec{j} & \vec{k} \\2 & 1 & -3 \\4 & 0 & 5\end{vmatrix} \\& = \vec{i} \begin{vmatrix} 1 & -3 \\ 0 & 5 \end{vmatrix} - \vec{j} \begin{vmatrix} 2 & -3 \\ 4 & 5\end{vmatrix} + \vec{k} \begin{vmatrix} 2 & 1 \\ 4 & 0 \end{vmatrix} \\& = \vec{i} \begin{pmatrix} 1 \times 5 - 0 \times (-3)\end{pmatrix} - \vec{j} \begin{pmatrix} 2\times 5 - 4 \times (-3) \end{pmatrix} + \vec{k} \begin{pmatrix} 2\times 0 - 4 \times 1 \end{pmatrix} \\& = \vec{i}\begin{pmatrix} 5 - 0\end{pmatrix} - \vec{j} \begin{pmatrix} 10 - (-12)\end{pmatrix} + \vec{k} \begin{pmatrix} 0 - 4 \end{pmatrix} \\& = \vec{i} \begin{pmatrix} 5 \end{pmatrix} - \vec{j}\begin{pmatrix} 10+12 \end{pmatrix} + \vec{k} \begin{pmatrix} - 4 \end{pmatrix} \\\vec{u} \times \vec{v} & = 5 \vec{i} - 22 \vec{j} - 4 \vec{k}\end{aligned}\]That's our final answer, we can now state that \(\vec{u} \times \vec{v} = 5 \vec{i} - 22 \vec{j} - 4 \vec{k} \).

Exercise 1

1. Find \(\vec{a} \times \vec{b}\), given \(\vec{a} = \begin{pmatrix}2 \\ 3 \\ 4 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} 5 \\ -2 \\ 1 \end{pmatrix}\).
2. Find \(\vec{c}\times \vec{d}\), given \(\vec{c} = 3 \vec{i} + 5 \vec{j} - \vec{k}\) and \(\vec{d} = \vec{i} -4 \vec{j} +2 \vec{k}\).
3. Find \(\vec{u}\times \vec{v}\), given \(\vec{u} = 6 \vec{i} - 3 \vec{k}\) and \(\vec{v} = -2 \vec{i} + \vec{j} + 5 \vec{k}\).
4. Find \(\vec{a} \times \vec{b}\), given \(\vec{a} = \begin{pmatrix}-2 \\ 0 \\ 6 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} 4 \\ 1 \\ 3 \end{pmatrix}\).
5. Find \(\vec{m}\times \vec{n}\), given \(\vec{m} = -3\vec{i} + 5 \vec{j} - \vec{k}\) and \(\vec{n} = 4\vec{j} + 7 \vec{k}\).
6. Find \(\vec{u}\times \vec{v}\), given \(\vec{u} = \begin{pmatrix}8 \\ - 1 \\ -2 \end{pmatrix}\) and \(\vec{v} = \begin{pmatrix}-3 \\ - 2 \\ -4 \end{pmatrix}\).
7. Find \(\vec{a}\times \vec{b}\), given \(\vec{a} = 3 \vec{i} - 2 \vec{j}\) and \(\vec{b} = 5 \vec{i} + \vec{j}\).
8. Find \(\vec{c}\times \vec{d}\), given \(\vec{c} = \begin{pmatrix} 2 \\ 3 \\ 1 \end{pmatrix}\) and \(\vec{d} = \begin{pmatrix} 0 \\ 3 \\ 5 \end{pmatrix}\).

Note: this exercise can be downloaded as a worksheet to practice with: Worksheet 1

AREAS OF PARALLELOGRAMS & TRIANGLES

The vector product, or cross product, of two vectors can be used to calculate the area of a parallelogram as well as the area of a triangle.

Example

A parallelogram has sides lengths defined by the vectors \(\vec{a} = \begin{pmatrix} 6 \\ -1 \\ 3 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} 1 \\ 5 \\ 4 \end{pmatrix}\).

Find this parallelogram's area.

AREA of a TRIANGLE

Given a triangle, whose sides are defined by two vectors \(\vec{a}\) and \(\vec{b}\), its area is given by:\[\text{Area} = \frac{1}{2} \begin{vmatrix} \vec{a}\times \vec{b}\end{vmatrix}\]Note : this is half of the area of the parallelogram whose sides are \(vec{a}\) and \(\vec{b}\).

Example

A triangle has two sides defined by the vectors \(\vec{a} = \begin{pmatrix} -1 \\ 7 \\ 1 \end{pmatrix}\) and \(\vec{b} = \begin{pmatrix} 1 \\ 4 \\ 3 \end{pmatrix}\).

Find this triangle's area.

Exercise 2 (Exam Style Questions)

1. The vectors \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) satisfy the equation \(\vec{a}+\vec{b} + \vec{c} = \vec{0}\). Show that \(\vec{a}\times \vec{b} = \vec{b}\times \vec{c} = \vec{c}\times \vec{a}\).