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]

Guest Aug 20, 2020