### Instruction: When every vector on the line $y = \frac{5}{2} x + 4$ is projected onto a certain vector $\mathbf{w},$ the result is always the vector $\mathbf{p}.$  Find the vector $\mathbf{p}.$.### Response: I want to find a vector $\mathbf{w}$ that has the property that projecting any vector on the line $y = \frac{5}{2} x + 4$ onto it gives the same vector $\mathbf{p}.$
I notice that the line $y = \frac{5}{2} x + 4$ has slope $\frac{5}{2},$ which means that any vector on the line has the direction $\langle 2, 5 \rangle.$
I also know that the projection of a vector $\mathbf{u}$ onto another vector $\mathbf{v}$ is given by $\frac{\mathbf{u} \cdot \mathbf{v}}{\mathbf{v} \cdot \mathbf{v}} \mathbf{v}.$
If I want the projection to be the same for any vector on the line, then I need the numerator of the projection formula to be constant, which means that $\mathbf{u} \cdot \mathbf{v}$ must be constant for any $\mathbf{u}$ on the line.
This implies that $\mathbf{v}$ must be orthogonal to the direction of the line, or in other words, perpendicular to the vector $\langle 2, 5 \rangle.$
One way to find a vector perpendicular to $\langle 2, 5 \rangle$ is to take its negative reciprocal slope, which is $-\frac{2}{5}.$
Then a vector with this slope is $\langle 5, -2 \rangle.$
So I can take $\mathbf{w} = \langle 5, -2 \rangle$ as a possible vector for the projection.
I can check that any vector on the line $\mathbf{u} = \langle 2x, 5x + 4 \rangle$ has a constant dot product with $\mathbf{w}:$

\[\mathbf{u} \cdot \mathbf{w} = (2x)(5) + (5x + 4)(-2) = 10x - 10x - 8 = -8\]
This means that the projection of any vector on the line onto $\mathbf{w}$ is given by

\[\frac{\mathbf{u} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \mathbf{w} = \frac{-8}{29} \langle 5, -2 \rangle\]
This is the vector $\mathbf{p}$ that I am looking for.
It does not depend on the choice of $\mathbf{u}$ on the line, so it is the same for every vector on the line.