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

$\boxed{020}$ .

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2013 Mock AIME I Problems/Problem 8

Problem

Solution

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