The Cross Product
The cross product is the first vector that can only be used in the third dimension. In cartesian coordinates, we call the third dimension z, adding to the x and y dimensions that we are familiar with. Traditionally, z is vertical, x is horizontal, and y is forward and backward. Visualizing 3D mathematics can be rather difficult, so it can be easier to create all three planes (xy, yz, and xz) and use those individually.
The cross product of two vectors results in a third vector, unlike dot products which result in a value. The cross-product will result in a vector that is perpendicular to both vectors used.
V = {1, 2, 3}
W = {4, 5, 6}
The blue vector is the cross-product of the vectors X and Y. As you can see, it is perpendicular to both vectors. The mathematical approach to solving a cross-product is rather complicated. First, you have to create a 3 x 3 matrix with each vector and its corresponding dimension (i, j, and k are used to represent the x, y, and z dimensions). Then you take the determinant of that matrix to find the cross product. Here is how you solve that step by step (V x W is the way to show the cross product of V and W):
The geometric approach to finding the cross product is using this formula, where once again, Θ is the angle between the vectors,
V x W = |V| |W| Sin(Θ)
which interestingly enough finds the area of a parallelogram with vectors V and W as two sides. When you complete the parallelogram, the area can be found with Base times Height, which equals V x W
Furthermore, a plane can be created with only two things, a point {a, b, c} and a normal vector, V. A normal vector is simply a perpendicular vector to the plane. A normal vector looks like this:
Using these, we can create an equation for a plane with the equation
which simplifies to:
and finally:
Ax + By + Cz = D
where {A, B, C} is the vector perpendicular to the plane.