Cross product: Formula, Rule, And Examples

What is Cross product

Cross product is one of kind of vector product. It is also called vector product. It is because cross product is operated in vector case.

The result is length and direction. It is multiply two vectors (vector a and b) to get the result (length and the direction).

The length (magnitude) is describe the area of vectors (vector a and b). The shape is parallelogram and vector a and b as the sides.

In another word, if vector a and b has right angle then the length (magnitude) of the result of cross product vector a and b same as multiply the both of them.

The direction of result of cross product describe the direction as the product of vector a and b. It will be easier to determine the direction using right hand rule.

There is a case that make the result of cross product is zero. It is if two vector has same direction or the vectors have exactly opposite direction.

Another case is if length of one of vectors is zero. But if vectors a and b have right angle it will has maximum length.

Cross product formula

Cross product between two vectors is describe in three-dimensional space. It is symbolized as a x b.

Cross product formula is

|a| × |b| × sin(a) × n


  • a and b : vectors
  • α : angle between vector a and b (it is between 0°-180°)
  • n : unit vector at right angles to both a and b.

If there is vector a (xa, ya, za) and vector b (xb, yb, zb) that start from origin point (0,0,0) then the cross product is vector c (xc, yc, zc). Vector c is also can be determined by using :

xc = ya.zb – za.yb

yc = za.xb – xa.zb

zc = – ya.bx

Right hand rule

Right hand rule describe the cross product between two vectors (vector a and b) and the result (vector c). It is on three-dimensional space.

Specifically, right hand rule describe the direction of the result. Using index finger as vector a and middle finger as vector b then the direction is the thumb.


1. Vector a has magnitude 4, vector b  has magnitude 5, the angle between a and b is 30° and n is the unit vector at right angles to both a  and b. Determine a × b


Known that

|a| = 4

|b| = 5

α = 30o

Using cross product formula, we get

|a| x |b| x sin(a) x n = 4.5.sin(30˚) x n = 10n

2. What is the cross product of a (1,3,-2) and b (-2,1,3)? And determine the magnitude of the result.


xa = 1; ya = 3; za = -2

xb = -2; yb = 1; zb = 3

then if the result is vector c,

xc = 3.3 – (-2).1 = 9 +2 = 11

yc = (-2).(-2) – 1.3 = 4 – 3 = 1

zc = 1.1 – 3.(-2) = 1 + 6 = 7

so, the result (vector c) is (11,1,7)

the magnitude of vector c is

|c| = √112 + 12 +72 = √171 = 3√19