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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

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