Skip to main content

8. Cross Product (aka Vector Product) of 2 Vectors

Suppose we have 2 vectors A and B. These 2 vectors lie on a plane and the unit vector n is normal (at right angles) to that plane.

The cross product (also known as the vector product) of A and B is given by:

A × B = |A| |B| sin θ n

The right hand side represents a vector at right angles to the plane containing vectors A and B.

Note: Some textbooks use the following notation for the cross product: A∧B.

Example

In the earlier application involving a cubic box (see Vectors in 3D Application), we had a unit cube that had one corner at the origin. We found that the diagonal vectors BS and CP meet at an angle of `70.5^@` at the center of the cube.

Using the same unit cube, find the vector product of the vectors BS and CP.

Answer

vectors - 3D box example

Using the formula

A × B = |A| |B| sin θ n

and our values from before which were

|BS| = √3
|CP| = √3
θ = 70.5°

we have:

|BS| × |CP|

= |BS| |CP| sin θ n

= (√3)(√3) sin 70.5° n

= 2.828 n

This means our result is a vector with magnitude `2.828` units and direction perpendicular to the plane containing BS and CP, indicated in green in the following diagram.

final - cross product demonstration

Please support IntMath!

top

Search IntMath, blog and Forum

Search IntMath

Online Algebra Solver

This algebra solver can solve a wide range of math problems.

Math Lessons on DVD

Math videos by MathTutorDVD.com

Easy to understand math lessons on DVD. See samples before you commit.

More info: Math videos

The IntMath Newsletter

Sign up for the free IntMath Newsletter. Get math study tips, information, news and updates each fortnight. Join thousands of satisfied students, teachers and parents!


See the Interactive Mathematics spam guarantee.