2013 Mock AIME I Problems/Problem 8

Problem

Let $\textbf{u}=4\textbf{i}+3\textbf{j}$ and $\textbf{v}$ be two perpendicular vectors in the $x-y$ plane. If there are $n$ vectors $\textbf{r}_i$ for $i=1, 2, \ldots, n$ in the same plane having projections of $1$ and $2$ along $\textbf{u}$ and $\textbf{v}$ respectively, then find $\sum_{i=1}^{n}\|\textbf{r}_i\|^2.$ (Note: $\textbf{i}$ and $\textbf{j}$ are unit vectors such that $\textbf{i}=(1,0)$ and $\textbf{j}=(0,1)$ , and the projection of a vector $\textbf{a}$ onto $\textbf{b}$ is the length of the vector that is formed by the origin and the foot of the perpendicular of $\textbf{a}$ onto $\textbf{b}$ .)

Solution

Let $\theta$ be the angle that $u$ makes with the positive $x$ -axis. Now, rotate the plane clockwise by $\theta$ such that $u$ is on the new $x'$ -axis, perpendicular to the $y'$ -axis, on which $v$ now lies, as seen in the diagram below. Rotations do not affect the magnitude of the vectors $r_i$ , so our answer will be the same.

$[asy] import geometry; // x' and y' Axes draw((-8,0)--(8,0), arrow=Arrows); label("$x'$", (9,0)); draw((0,-8)--(0,8), arrow=Arrows); label("$y'$", (0,9)); // Vectors draw((0,0)--(5,0), red+linewidth(1.5), arrow=Arrow(TeXHead)); label("$u$", midpoint((0,0)--(5,0)), S); draw((0,0)--(0,6), blue+linewidth(1.5), arrow=Arrow(TeXHead)); label("$v$", midpoint((0,0)--(0,6)), W); // Right Angle Mark markscalefactor = 0.12; draw(rightanglemark((1,0),(0,0),(0,1))); [/asy]$

Now, from the information given by the problem, it is clear that the rotated vectors $r_i'$ are those with coordinates $(\pm1, \pm2)$ . Thus, there are $4$ of these vectors, each with a squared magnitude of $(\pm 1)^2+(\pm 2)^2 = 1+4 = 5$ . Thus, our answer is $4 \cdot 5 = \boxed{020}$ .

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2013 Mock AIME I Problems/Problem 8

Problem

Solution

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