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Suppose we have a vector $a \in \mathcal{R}^2$ and $x,y \in \mathcal{R}^2$ and let $\alpha$ is the angle between $x$ and $y$. Define: $$z = (a \cdot x) x + (a \cdot y)y$$

If $\alpha = \pi /2$ then $z = x$, if $\alpha < \pi /2$ $z$ is longer than $x$ otherwise it is shorter.

How is the length of $z$ related with $\alpha$? Does it depends also on the relative angle between $x$ and $a$ or $y$ and $a$?

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