Precalculus (6th Edition) Blitzer

Any two vectors $\mathbf{v}$ and $\mathbf{w}$ in the same direction will satisfy the condition that projection of $\mathbf{v}$ onto $\mathbf{w}$ is $\mathbf{v}$.
The projection of $\mathbf{v}$ onto $\mathbf{w}$ is expressed as follows: $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}=\frac{\mathbf{v}\centerdot \mathbf{w}}{{{\left\| \mathbf{w} \right\|}^{2}}}\mathbf{w}$ Substituting the value of $\text{pro}{{\text{j}}_{\mathbf{w}}}\mathbf{v}$ as $\mathbf{v}$ from equation (1), we get $\mathbf{v}=\frac{\mathbf{v}\centerdot \mathbf{w}}{{{\left\| \mathbf{w} \right\|}^{2}}}\mathbf{w}$ (2) Now, $\frac{\mathbf{v}\centerdot \mathbf{w}}{{{\left\| \mathbf{w} \right\|}^{2}}}$ is a scalar quantity. Therefore, consider $\frac{\mathbf{v}\centerdot \mathbf{w}}{{{\left\| \mathbf{w} \right\|}^{2}}}=k$ (3) Substituting equation (3) into equation (2), we get $\mathbf{v}=k\mathbf{w}$ When one vector is expressed as a scalar multiple of another, then those two vectors are in the same direction. Hence, two vectors $\mathbf{v}$ and $\mathbf{w}$ moving in the same direction will satisfy the condition that projection of $\mathbf{v}$ onto $\mathbf{w}$ is $\mathbf{v}$.