How Do You Calculate the Vector (Cross) Product of Two Vectors?

The vector product, or cross product, of two three-dimensional vectors is found by combining their components in a fixed order. The result is a vector perpendicular to both original vectors, and its direction follows the right-hand screw rule.

Calculation Method

For vectors $\mathbf{a}=(a_1,a_2,a_3)$ and $\mathbf{b}=(b_1,b_2,b_3)$, use

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

The component order is first $2,3$, then $3,1$, then $1,2$. A reliable alternative is determinant expansion, but the middle component must carry the correct negative sign.

Let $\mathbf{a}=(1,2,3)$ and $\mathbf{b}=(4,-1,2)$. Then

$$\mathbf{a}\times\mathbf{b}=(2(2)-3(-1),\ 3(4)-1(2),\ 1(-1)-2(4))=(7,10,-9).$$

Check perpendicularity using scalar products:

$$(7,10,-9)\cdot(1,2,3)=7+20-27=0,$$ $$(7,10,-9)\cdot(4,-1,2)=28-10-18=0.$$

Both results are zero, so the cross product is orthogonal to each vector.

Direction and Orientation

The cross product has two possible perpendicular directions. Use the right-hand screw rule: point your right-hand fingers along $\mathbf{a}$, curl them toward $\mathbf{b}$ through the smaller angle, and your thumb indicates $\mathbf{a}\times\mathbf{b}$. Swapping the vectors reverses this direction.

If asked for a normal vector, any non-zero scalar multiple of the cross product is valid. A unit normal requires division by its magnitude.

Geometric Meaning

The magnitude satisfies

$$|\mathbf{a}\times\mathbf{b}|=|\mathbf{a}||\mathbf{b}|\sin\theta.$$

It gives the parallelogram's area. Here,

$$|\mathbf{a}\times\mathbf{b}|=\sqrt{7^2+10^2+(-9)^2}=\sqrt{230},$$

so the corresponding triangle has area $\frac{1}{2}\sqrt{230}$. The unit normal is $\frac{}{}$.

This also explains why parallel vectors have zero cross product: the angle between them is $0$ or $\pi$, so $\sin\theta=0$. If the cross product is zero for non-zero vectors, they are parallel or anti-parallel. This is a useful geometric check on your calculation.

A common misconception is that the cross product is a scalar. It is a vector; the dot product produces a scalar. Triangle area is half the cross-product magnitude.

IB Exam Technique

This is HL-only content from AHL 3.13: Scalar and vector products. All Mathematics AI HL papers are GDC-active, but for calculate show the component substitution before stating the final vector. For area questions, take the magnitude and halve it only for a triangle.