Vector Projections

The final part of the vector operations section involves using concepts of dot product and orthogonality to compute vector projections. Given two vectors u and v, the vector projection (or projection) of v on u is denoted by projuv

The term projection comes from the idea of shining a light straight down on vector u and seeing the shadow cast by v on u. This 'shadow' is in the same direction as u, and may have a different length. Therefore, projuv is a scalar multiple of u. The projection is found by the formula,

Since the numerator of our scalar is the dot product u • v, we can see that the projection is the zero vector if u and v are orthogonal. This fits with our analogy, as a perpendicular vector v would not cast any shadow on u. Similarly, if u = v, then of course projuv = u, since ||u||2 = u • u, so our scalar would equal 1. This again fits with our shadow analogy, which is illustrated below,

The component of v on u (or scalar projection) is equal to the length of the vector projection of v on u. It is denoted compuv, and is given by the formula,

Let's look at a few examples involving projections and components