What Is the Vector Product?

The vector product of the vectors u and v

The Vector product (cross product) is a new way to multiply vectors, where the answer is also a vector! I think the absolute easiest way to calculate the cross product is to cross in towers:

Rule

The Cross Product of Two Vectors

u ×v = (x1, y1, z1) × ( x 2, y2, z2) = (y1z2 y2z1,z1x2 z2x1, x1y2 x2y1)

u×v = (x1, y1, z1) × ( x 2, y2, z2) = (y1z2 y2z1,z1x2 z2x1,x1y2 x2y1)

Don’t try to remember the letter combinations, that is really cumbersome and unnecessary. Rather, try to remember the pattern you follow when crossing the vectors. You will discover that y1z2 y2z1 forms a cross when you multiply like in the formula:

( y1, z1) × (y 2, z2)

The same goes for z1x2 z2x1:

(x1, z1) × (x 2, z2)

And it also goes for x1y2 x2y1:

(x1, y1, ) × (x 2, y2, )

Notice that the next cross begins with the variable that the last cross ended with. That means you only need to remember to begin in the middle and move towards the right. Make up two vectors and try the method out, you won’t regret it!

Rule

The cross product u ×v is perpendicular to both u and v.

If you use this formula, you can find the length of a cross product if you know the angle α between the two vectors:

Formula

Length of the Cross Product

|u ×v| = |u| |v| sin α,α [0°, 180°)

If you have the cross product as a vector, you can use the formula for vector length.

Example 1

Find the cross product of u = (1, 3,2) and v = (3, 2, 4). You get

u ×v = ( 1, 3, 2) × ( 3, 2, 4) = (3 4 2 (2), (2) (3) 1 4, 1 2 (3) 3) = (12 + 4, 6 4, 2 + 9) = (16, 2, 11) .

u ×v = ( 1, 3, 2) × ( 3, 2, 4) = (3 4 2 (2), (2) (3) 1 4, 1 2 (3) 3) = (12 + 4, 6 4, 2 + 9) = (16, 2, 11) .

Example 2

You have been given the vectors u and v, and the angle between them is α = 30°. You know that |u| = 3 and |v| = 5. How long is the cross product of the two vectors?

You insert the numbers into the formula, which gives you

|u ×v| = |u| |v| sin 30° = 3 5 0.5 = 7.5.

|u ×v| = |u| |v| sin 30° = 3 5 0.5 = 7.5.

That means u ×v will have a length of 7.5.

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