Consider the projection matrix $\mathbf{P}$ that satisfies
\[\mathbf{P} \mathbf{v} = \text{Projection of $\mathbf{v}$ onto } \begin{pmatrix} 4 \\ 1 \end{pmatrix}.\]
Use the picture below to find vectors with integer components and magnitude no greater than $8$ that satisfy
\[\mathbf{P} \mathbf{v} = \begin{pmatrix} 4 \\1 \end{pmatrix}.\]
List your answers as columns in increasing order of -component.
[asy]
size(200);
import olympiad;
//Draws grid lines with the given parameters:
void gridLines(int xmin, int xmax, int ymin, int ymax)
{
for(int i = ymin+1; i< ymax; ++i)
{
draw((xmin,i)--(xmax,i), mediumgray);
}
for(int i = xmin+1; i< xmax; ++i)
{
draw((i,ymax)--(i,ymin), mediumgray);
}
}
//Gives the maximum line that fits in the box.
path maxLine(pair A, pair B, real xmin, real xmax, real ymin, real ymax)
{
pair[] endpoints = intersectionpoints(A+10(B-A) -- A-10(B-A), (xmin, ymin)--(xmin, ymax)--(xmax, ymax)--(xmax, ymin)--cycle);
if (endpoints.length >= 2) return endpoints[0]--endpoints[1];
else return nullpath;
}
pair U,V, P;
U = (4,1);
V = (5,-3);
P = foot(V, U, (0, 0));
gridLines(-9, 8, -6,6);
draw(maxLine((0,0), U,-9, 8, -6,6));
draw(-2U-- -U, red, Arrow(size = 0.3cm));
draw((0, 0)--P, heavygreen, Arrow(size = 0.3cm));
dot(-2U, red);
dot((0,0), heavygreen);
//draw(maxLine(P, V,-8, 6, -6,6), dashed);
//draw(rightanglemark(2U, P, V, 10));
label("$\mathbf{u}$", -3U/2, S);
//label("$\mathbf{v}$",V/2,SW);
[/asy]
