2.3 The dot product  (Page 5/16)

Projections

As we have seen, addition combines two vectors to create a resultant vector. But what if we are given a vector and we need to find its component parts? We use vector projections to perform the opposite process; they can break down a vector into its components. The magnitude of a vector projection is a scalar projection. For example, if a child is pulling the handle of a wagon at a 55° angle, we can use projections to determine how much of the force on the handle is actually moving the wagon forward ( [link] ). We return to this example and learn how to solve it after we see how to calculate projections.

Sometimes it is useful to decompose vectors—that is, to break a vector apart into a sum. This process is called the resolution of a vector into components . Projections allow us to identify two orthogonal vectors having a desired sum. For example, let $\text{v}=⟨6,\mathrm{-4}⟩$ and let $\text{u}=⟨3,1⟩.$ We want to decompose the vector v into orthogonal components such that one of the component vectors has the same direction as u .

We first find the component that has the same direction as u by projecting v onto u . Let $\text{p}={\text{proj}}_{\text{u}}\text{v}.$ Then, we have

Now consider the vector $\text{q}=\text{v}-\text{p}.$ We have

Clearly, by the way we defined q , we have $\text{v}=\text{q}+\text{p},$ and