Answer
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 $\mathbf{v}$.
Work Step by Step
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 $\mathbf{v}$.
