## Precalculus (6th Edition) Blitzer

The projection of a vector $\mathbf{v}$ onto a vector $\mathbf{w}$ is $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=\frac{\mathbf{v}\centerdot \mathbf{w}}{{{\left\| \mathbf{w} \right\|}^{\mathbf{2}}}}\mathbf{w}$. The projection is shown below:
First, draw the vectors $\mathbf{v}$ and $\mathbf{w}$ with the same initial point in two different directions. Suppose the angle between $\mathbf{v}$ and $\mathbf{w}$ is $\theta$. Now, draw a perpendicular line to the vector $\mathbf{w}$ from the vertex of $\mathbf{v}$. The line joining in the direction of the perpendicular line of $\mathbf{v}$ onto the $\mathbf{w}$ from initial point is represented as $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}$. Consider the cosine function to get $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}$. \begin{align} & \cos \theta =\frac{\left\| \text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v} \right\|}{\left\| \mathbf{v} \right\|} \\ & \left\| \text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v} \right\|=\left\| \mathbf{v} \right\|\cos \theta \end{align} Consider, $\left\| \text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v} \right\|=\left\| \mathbf{v} \right\|\cos \theta$ $\cdot \cdot \cdot \cdot \cdot \cdot$ (1) The projection of vectors is also represented as the dot product. Hence, the dot product of vectors $\mathbf{v}$ and $\mathbf{w}$ is represented as \begin{align} & \mathbf{v}\centerdot \mathbf{w}=\left\| \mathbf{v} \right\|\left\| \mathbf{w} \right\|\text{cos }\!\!\theta\!\!\text{ } \\ & \frac{\mathbf{v}\centerdot \mathbf{w}}{\left\| \mathbf{w} \right\|}=\left\| \mathbf{v} \right\|\text{cos }\!\!\theta\!\!\text{ } \end{align} Consider, $\frac{\mathbf{v}.\mathbf{w}}{\left\| \mathbf{w} \right\|}=\left\| \mathbf{v} \right\|\text{cos }\!\!\theta\!\!\text{ }$ $\cdot \cdot \cdot \cdot \cdot \cdot$ (2) Compare equations (1) and (2) and get $\left\| \text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v} \right\|=\frac{\mathbf{v}\centerdot \mathbf{w}}{\left\| \mathbf{w} \right\|}$ The above expression is used to find the magnitude of a projection vector. Now get the projection vector using a unit vector of $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}$ in the direction of the vector $\mathbf{w}$. \begin{align} & \text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=\left( \frac{\mathbf{v}\centerdot \mathbf{w}}{\left\| \mathbf{w} \right\|} \right)\left( \frac{\mathbf{w}}{\left\| \mathbf{w} \right\|} \right) \\ & \text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=\frac{\mathbf{v}\centerdot \mathbf{w}}{{{\left\| \mathbf{w} \right\|}^{2}}}\mathbf{w} \\ \end{align}