Dot Product continued

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Projections

One of the main applications of dot product is in calculating a projection of one vector onto another

The two important formulas are:

\[\begin{aligned} \operatorname{comp}_{\mathbf{b}}\mathbf{a} &= \frac{\mathbf{a}\cdot\mathbf{b}}{\left\lvert\mathbf{b} \right\rvert}\\ \operatorname{proj}_{\mathbf{b}}\mathbf{a} &= \frac{\mathbf{a}\cdot\mathbf{b}}{\mathbf{b}\cdot\mathbf{b}}\mathbf{b} \end{aligned}\]

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Direction cosines

One way to describe the direction of a vector \(\mathbf{v}\) in 3d is by giving a unit vector with the same direction as \(\mathbf{v}\). What is the actual meaning of its components?

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