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Imagine finding a new direction that is perpendicular to two directions you already know. That might sound complicated, but it is exactly what the cross product of two vectors allows you to do. The interesting part is that a small change in the order of your calculation can reverse the direction of your answer—even when the numbers look almost identical.
So, how do you calculate it correctly, and how can you tell whether your answer makes sense?
What Is the Cross Product of Two Vectors?
The cross product is an operation that takes two three-dimensional vectors and produces a third vector perpendicular to both. It is written as A × B and is also called the vector product.
Unlike the dot product, which produces a single number, the cross product produces a vector with three components. Those components tell you the direction and magnitude of the result.
A × B = (Cₓ, Cᵧ, Cᵤ)
But getting those components right depends on following a particular pattern.
How to Calculate the Cross Product Step by Step
Suppose we have two vectors:
Vector A
A = (2, −1, 3)
Vector B
B = (−3, 4, 1)
To find their cross product, use this formula:
A × B = (AᵧBᵤ − AᵤBᵧ, AᵤBₓ − AₓBᵤ, AₓBᵧ − AᵧBₓ)
Here, the third coordinate should be written as z, so the standard formula is:
A × B = (AᵧBᶻ − AᶻBᵧ, AᶻBₓ − AₓBᶻ, AₓBᵧ − AᵧBₓ)
Each component follows the same basic idea: multiply two pairs of coordinates and subtract the products. Let's work through the example.
Step 1: Find the first component
Use the y- and z-coordinates of both vectors:
(−1 × 1) − (3 × 4)
= −1 − 12
= −13
Step 2: Find the second component
Now use the z- and x-coordinates:
(3 × −3) − (2 × 1)
= −9 − 2
= −11
Notice something important: the order of subtraction matters. Swapping the two products would change the sign of this component.
Step 3: Find the third component
Use the x- and y-coordinates:
(2 × 4) − (−1 × −3)
= 8 − 3
= 5
Put the three results together:
A × B = (−13, −11, 5)
That's the cross product. But before moving on, there is a useful question worth asking: how do we know these three numbers really give us a vector perpendicular to both original vectors?
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How to Check Whether Your Answer Is Correct
One of the best ways to check a cross product is to use the dot product. Two perpendicular vectors have a dot product of zero.
Our result is C = (−13, −11, 5).
First, check it against A:
A · C = 2(−13) + (−1)(−11) + 3(5)
= −26 + 11 + 15
= 0
Now check it against B:
B · C = (−3)(−13) + 4(−11) + 1(5)
= 39 − 44 + 5
= 0
Both results equal zero. That confirms that C is perpendicular to both A and B.
This is more than a way to check homework. It is a way to understand what makes the cross product different from an ordinary multiplication result.
What Does the Cross Product Tell You?
The calculation gives you a vector, but its magnitude and direction reveal additional information.
Direction
The resulting vector is perpendicular to both original vectors. The right-hand rule determines which of the two possible perpendicular directions it points in.
Magnitude
The length of the cross product is |A × B| = |A||B| sin θ, where θ is the angle between the vectors.
Area
The magnitude of the cross product equals the area of the parallelogram formed by the two vectors.
Here is an interesting consequence: if two vectors are parallel, their cross product is the zero vector. This happens because the sine of the angle between them is zero when the angle is 0° or 180°.
And remember the order sensitivity mentioned at the beginning? Reversing the vectors reverses the result:
B × A = −(A × B)
The magnitude stays the same, but the direction changes. The cross product is therefore not commutative, unlike ordinary multiplication.
Frequently Asked Questions
1. What is the difference between the cross product and the dot product?
The cross product of two three-dimensional vectors produces another vector perpendicular to both. The dot product produces a scalar, or single number, and is calculated by multiplying corresponding components and adding the results.
2. Can you find the cross product of two 2D vectors?
The standard cross product is defined for three-dimensional vectors. You can represent two-dimensional vectors in 3D by adding a zero z-coordinate. The resulting cross product will have a zero x- and y-component, with its z-component indicating the signed magnitude of the 2D cross product.
3. Why does the cross product of parallel vectors equal zero?
The magnitude formula is |A × B| = |A||B| sin θ. For parallel or oppositely directed vectors, θ is 0° or 180°, so sin θ = 0. The resulting vector therefore has zero magnitude.
4. Does changing the order of the vectors change the cross product?
Yes. Reversing the order changes the sign of the result: B × A = −(A × B). The two results have the same magnitude but point in opposite directions.
