How do I find the vector/cross product of two three-dimensional vectors?

Finding the cross product of two 3D vectors uses a very similar method to that of finding the determinant of a 3x3 matrix.

So, to find the cross product, just follow these simple steps:

1. To begin, create a 3x3 matrix with the unit vectors i, j and k filling the top row.
2. Take the first vector and put the x component under the i, the y component under the j and the z component under the k. Repeat this for the second vector by putting the x, y and z components in their correct columns in the bottom row of the matrix.
3. Next, cover the first row and the first column (the row and column containing the i unit vector). This leaves a 2x2 matrix still showing. You should find the determinant of this matrix (remember from FP1: determinant = ad-bc). This determinant then becomes the x component of the final vector we are trying to find.
4. Now uncover the first row and column and cover the row and column containing the j unit vector. Find the determinant of the resulting 2x2 matrix and then multiply the determinant by -1. This becomes the y component of the final vector.
5. Repeat this for the k unit vector, but without multiplying by -1, to find the z component of the final vector.
6. You now have the final vector - this is the vector product of your original vectors!

Remember: You only have to multiply the y component (when you cover the j) by -1!