Cross Product — Formula, Direction & Applications

The cross product takes two vectors in 3D and returns a new vector perpendicular to both. Its size is

$$|a \times b| = |a||b|\sin\theta$$

where $\theta$ is the angle between $a$ and $b$, and in a right-handed coordinate system its direction follows the right-hand rule.

This gives the main idea quickly. Parallel vectors have $\sin\theta = 0$, so their cross product is the zero vector. Perpendicular vectors have $\sin\theta = 1$, so the cross product has the largest possible magnitude for those vector lengths.

Cross Product Formula In Coordinates

If

$$a = (a_1, a_2, a_3), \qquad b = (b_1, b_2, b_3)$$

then

$$a \times b = (a_2b_3 - a_3b_2,\ a_3b_1 - a_1b_3,\ a_1b_2 - a_2b_1)$$

The result is a vector, not a scalar. That is one of the main differences from the dot product.

Cross Product Direction And The Right-Hand Rule

The cross product points perpendicular to the plane containing $a$ and $b$. In a right-handed coordinate system, curl the fingers of your right hand from $a$ toward $b$ through the smaller angle. Your thumb points in the direction of $a \times b$.

Order matters:

$$a \times b = -(b \times a)$$

So swapping the vectors reverses the direction.

Worked Example: Find $a \times b$

Take

$$a = (2, 0, 0), \qquad b = (0, 3, 0)$$

Using the component formula,

$$a \times b = (0 \cdot 0 - 0 \cdot 3,\ 0 \cdot 0 - 2 \cdot 0,\ 2 \cdot 3 - 0 \cdot 0)$$

so

$$a \times b = (0, 0, 6)$$

This example is useful because everything is visible at once:

• The result points in the positive $z$ direction, so it is perpendicular to both input vectors.
• Its magnitude is $6$.
• That same magnitude is the area of the parallelogram formed by $a$ and $b$.

You can check the magnitude formula too. Here $|a| = 2$, $|b| = 3$, and the angle is $90^\circ$, so

$$|a \times b| = 2 \cdot 3 \cdot \sin 90^\circ = 6$$

If you reverse the order, then

$$b \times a = (0, 0, -6)$$

The size stays the same, but the direction flips.

What The Cross Product Magnitude Means

The quantity $|a \times b|$ gives the area of the parallelogram spanned by $a$ and $b$. If you want the area of the triangle formed by the same vectors, divide by $2$.

That geometric meaning explains why the cross product becomes zero for parallel vectors: a parallelogram with no width has area zero.

Common Cross Product Mistakes

Mixing It Up With The Dot Product

The dot product gives a scalar and uses $\cos\theta$. The cross product gives a vector and uses $\sin\theta$ for its magnitude.

Forgetting That Order Matters

$a \times b$ and $b \times a$ do not point in the same direction. They are negatives of each other.

Using It Outside The Usual 3D Setting

In most school and first-course engineering contexts, the cross product is defined for two 3D vectors. If you are working in 2D, people often switch to scalar-area interpretations or embed the vectors into 3D first.

Where The Cross Product Is Used

In geometry, the cross product helps find areas and normal vectors. In vector calculus and graphics, it is used to build a perpendicular direction to a surface or plane.

In physics, it appears when both magnitude and rotational direction matter. A standard example is torque:

$$\tau = r \times F$$

That formula says the turning effect depends on the position vector, the force, and the angle between them.

Try A Similar Problem

Try

$$a = (1, 1, 0), \qquad b = (2, -1, 0)$$

Compute $a \times b$, then check its magnitude against $|a||b|\sin\theta$. After that, reverse the order and confirm that the new answer points in the opposite direction.