What is a Vector?

What is a Cross Product?

Formula for Cross Product

Steps to Calculate the Cross Product

Properties of the Cross Product

Magnitude of the Cross Product

Applications of Cross Product

Geometric Interpretation

Examples

FAQs on Cross Product

Back to Blog

Important Topic of Maths: Cross Product

In mathematics, the cross product is a way to multiply two vectors in three-dimensional space to get another vector that is perpendicular to both of the original vectors. The cross product is a fundamental concept in physics, engineering, and computer science. Let's break it down step by step and understand it in simple terms.

What is a Vector?

Before we dive into the cross product, let’s quickly recap what a vector is. A vector is a quantity that has both a direction and a magnitude (size). For example:

speak to our academic counsellor

access to India's best teachers with a record of producing top rankers year on year.

+91

The velocity of a moving car (it tells us how fast and in which direction the car is moving).
The force acting on an object (it has a strength and a direction).

In three-dimensional space, a vector can be written as:

\mathbf{A} = (a_1, a_2, a_3)

$A$ $=$ $($ $a$ $1$ $$ $,$ $a$ $2$ $$ $,$ $a$ $3$ $$ $)$

Unlock the full solution & master the concept

Get a detailed solution and exclusive access to our masterclass to ensure you never miss a concept

Here, $a_1, a_2,$ $a$ $1$ $$ $,$ $a$ $2$ $$ $,$ and $a_3$ $a$ $3$ $$ are the components of the vector along the x, y, and z axes.

What is a Cross Product?

The cross product is a mathematical operation that takes two vectors as input and produces another vector as output. This resulting vector is:

Ready to Test Your Skills?

Check Your Performance Today with our Free Mock Tests used by Toppers!

Take Free Test

Perpendicular to both the input vectors.
Its magnitude depends on the angle between the input vectors and their magnitudes.

The cross product is only defined for vectors in three dimensions.

Formula for Cross Product

If we have two vectors:

create your own test

YOUR TOPIC, YOUR DIFFICULTY, YOUR PACE

start learning for free

\mathbf{A} = (a_1, a_2, a_3) \quad \text{and} \quad \mathbf{B} = (b_1, b_2, b_3)

$A$ $=$ $($ $a$ $1$ $$ $,$ $a$ $2$ $$ $,$ $a$ $3$ $$ $)$ $and$ $B$ $=$ $($ $b$ $1$ $$ $,$ $b$ $2$ $$ $,$ $b$ $3$ $$ $)$

The cross product, written as $\mathbf{A} \times \mathbf{B}$ $A$ $\times$ $B$ , is calculated using the following formula:

\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{vmatrix}

$A$ $\times$ $B$ $=$ $$ $i$ $a$ $1$ $$ $b$ $1$ $$ $$ $j$ $a$ $2$ $$ $b$ $2$ $$ $$ $k$ $a$ $3$ $$ $b$ $3$ $$ $$ $$

Here, $\mathbf{i}$ $i$ , $\mathbf{j}$ $j$ , and $\mathbf{k}$ $k$ are the unit vectors along the x, y, and z axes. The determinant expands to:

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

$A$ $\times$ $B$ $=$ $i$ $($ $a$ $2$ $$ $b$ $3$ $$ $-$ $a$ $3$ $$ $b$ $2$ $$ $)$ $-$ $j$ $($ $a$ $1$ $$ $b$ $3$ $$ $-$ $a$ $3$ $$ $b$ $1$ $$ $)$ $+$ $k$ $($ $a$ $1$ $$ $b$ $2$ $$ $-$ $a$ $2$ $$ $b$ $1$ $$ $)$

This gives us the components of the resulting vector.

Steps to Calculate the Cross Product

Write down the vectors: Let’s say $\mathbf{A} = (2, 3, 4)$ $A$ $=$ $($ $2$ $,$ $3$ $,$ $4$ $)$ and $\mathbf{B} = (5, 6, 7)$ $B$ $=$ $($ $5$ $,$ $6$ $,$ $7$ $)$ .
Set up the determinant:

$\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 2 & 3 & 4 \\ 5 & 6 & 7 \\ \end{vmatrix}$ $A$ $\times$ $B$ $=$ $$ $i$ $25$ $$ $j$ $36$ $$ $k$ $47$ $$ $$

Expand the determinant:

$\mathbf{A} \times \mathbf{B} = \mathbf{i}((3)(7) - (4)(6)) - \mathbf{j}((2)(7) - (4)(5)) + \mathbf{k}((2)(6) - (3)(5))$ $A$ $\times$ $B$ $=$ $i$ $(($ $3$ $)$ $($ $7$ $)$ $-$ $($ $4$ $)$ $($ $6$ $))$ $-$ $j$ $(($ $2$ $)$ $($ $7$ $)$ $-$ $($ $4$ $)$ $($ $5$ $))$ $+$ $k$ $(($ $2$ $)$ $($ $6$ $)$ $-$ $($ $3$ $)$ $($ $5$ $))$

Simplify:

$\mathbf{A} \times \mathbf{B} = \mathbf{i}(21 - 24) - \mathbf{j}(14 - 20) + \mathbf{k}(12 - 15)$ $A$ $\times$ $B$ $=$ $i$ $($ $21$ $-$ $24$ $)$ $-$ $j$ $($ $14$ $-$ $20$ $)$ $+$ $k$ $($ $12$ $-$ $15$ $)$ $\mathbf{A} \times \mathbf{B} = \mathbf{i}(-3) - \mathbf{j}(-6) + \mathbf{k}(-3)$ $A$ $\times$ $B$ $=$ $i$ $($ $-3$ $)$ $-$ $j$ $($ $-6$ $)$ $+$ $k$ $($ $-3$ $)$

Combine the components:

$\mathbf{A} \times \mathbf{B} = (-3, 6, -3)$ $A$ $\times$ $B$ $=$ $($ $-3$ $,$ $6$ $,$ $-3$ $)$

So, the resulting vector is $(-3, 6, -3)$ $($ $-3$ $,$ $6$ $,$ $-3$ $)$ .

Properties of the Cross Product

Perpendicular Vector: The result of the cross product is always a vector that is perpendicular to both the input vectors. This means it is at a right angle to the plane formed by the two vectors.
Not Commutative: The cross product is not commutative, meaning:

$\mathbf{A} \times \mathbf{B} \neq \mathbf{B} \times \mathbf{A}$ $A$ $\times$ $B$ $\neq$ $=$ $B$ $\times$ $A$

In fact, $\mathbf{B} \times \mathbf{A} = -(\mathbf{A} \times \mathbf{B})$ $B$ $\times$ $A$ $=$ $-$ $($ $A$ $\times$ $B$ $)$ .

Zero Vector: If the two vectors are parallel or one of the vectors is a zero vector, the cross product is zero.

$\mathbf{A} \times \mathbf{A} = 0$ $A$ $\times$ $A$ $=$ $0$

Distributive Property: The cross product follows the distributive property:

$\mathbf{A} \times (\mathbf{B} + \mathbf{C}) = (\mathbf{A} \times \mathbf{B}) + (\mathbf{A} \times \mathbf{C})$ $A$ $\times$ $($ $B$ $+$ $C$ $)$ $=$ $($ $A$ $\times$ $B$ $)$ $+$ $($ $A$ $\times$ $C$ $)$

Magnitude of the Cross Product

The magnitude (size) of the resulting vector from the cross product is given by:

|\mathbf{A} \times \mathbf{B}| = |\mathbf{A}||\mathbf{B}|\sin\theta

$∣$ $A$ $\times$ $B$ $∣$ $=$ $∣$ $A$ $∣∣$ $B$ $∣$ $sin$ $θ$

Where:

$|\mathbf{A}|$ $∣$ $A$ $∣$ and $|\mathbf{B}|$ $∣$ $B$ $∣$ are the magnitudes of the vectors $\mathbf{A}$ $A$ and $\mathbf{B}$ $B$ .
$\theta$ $θ$ is the angle between the two vectors.

The magnitude tells us the area of the parallelogram formed by the two vectors.

Applications of Cross Product

The cross product is widely used in various fields. Here are some practical applications:

1. Physics:

Torque: Torque ( $\mathbf{\tau}$ $τ$ ) is calculated using the cross product of the force vector ( $\mathbf{F}$ $F$ ) and the position vector ( $\mathbf{r}$ $r$ ): $\mathbf{\tau} = \mathbf{r} \times \mathbf{F}$ $τ$ $=$ $r$ $\times$ $F$
Angular Momentum: Angular momentum is also determined using the cross product.

2. Engineering:

In mechanical engineering, the cross product helps in analyzing rotational forces.
In computer graphics, it helps calculate normals to surfaces for rendering 3D objects.

3. Mathematics:

Cross product is used to find the perpendicular vector to a plane defined by two vectors.

4. Navigation:

In navigation and aviation, cross products are used to calculate directions and solve vector-related problems.

Geometric Interpretation

The cross product has a very intuitive geometric meaning:

The direction of the resulting vector is determined by the right-hand rule:
- Point your fingers in the direction of the first vector ( $\mathbf{A}$ $A$ ).
- Curl them towards the second vector ( $\mathbf{B}$ $B$ ).
- Your thumb points in the direction of $\mathbf{A} \times \mathbf{B}$ $A$ $\times$ $B$ .
The magnitude of the cross product represents the area of the parallelogram formed by the two vectors.

Examples

Example 1:

\mathbf{A} = (1, 0, 0), \quad \mathbf{B} = (0, 1, 0)

$A$ $=$ $($ $1$ $,$ $0$ $,$ $0$ $)$ $,$ $B$ $=$ $($ $0$ $,$ $1$ $,$ $0$ $)$ $\mathbf{A} \times \mathbf{B} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & 0 \\ 0 & 1 & 0 \\ \end{vmatrix}$ $A$ $\times$ $B$ $=$ $$ $i$ $10$ $$ $j$ $01$ $$ $k$ $00$ $$ $$ $\mathbf{A} \times \mathbf{B} = \mathbf{i}(0 - 0) - \mathbf{j}(0 - 0) + \mathbf{k}(1 - 0)$ $A$ $\times$ $B$ $=$ $i$ $($ $0$ $-$ $0$ $)$ $-$ $j$ $($ $0$ $-$ $0$ $)$ $+$ $k$ $($ $1$ $-$ $0$ $)$ $\mathbf{A} \times \mathbf{B} = (0, 0, 1)$ $A$ $\times$ $B$ $=$ $($ $0$ $,$ $0$ $,$ $1$ $)$

The result is $(0, 0, 1)$ $($ $0$ $,$ $0$ $,$ $1$ $)$ , a vector along the z-axis.

Example 2:

\mathbf{A} = (2, 3, 4), \quad \mathbf{B} = (5, 6, 7)

$A$ $=$ $($ $2$ $,$ $3$ $,$ $4$ $)$ $,$ $B$ $=$ $($ $5$ $,$ $6$ $,$ $7$ $)$

We already calculated this:

\mathbf{A} \times \mathbf{B} = (-3, 6, -3)

$A$ $\times$ $B$ $=$ $($ $-3$ $,$ $6$ $,$ $-3$ $)$

FAQs on Cross Product

What means cross product?

Noun Mathematics. A vector perpendicular to two given vectors, u and v, and having value identical to the manufactured from the magnitudes of the two given vectors accelerated through the sine of the attitude among the two given vectors, commonly represented by means of u × v.

What is the cross result of two vectors?

Cross product of vectors is the technique of multiplication of two vectors. A pass product is denoted via the multiplication signal(x) between vectors. It is a binary vector operation, described in a three-dimensional gadget.

What is a Vector?

What is a Cross Product?

Formula for Cross Product

Steps to Calculate the Cross Product

Properties of the Cross Product

Magnitude of the Cross Product

Applications of Cross Product

Geometric Interpretation

Examples

FAQs on Cross Product