When projection of $a$ onto $b$ is equal to projection of $ b$ onto $a$

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Having a lot of trouble with my Linear Algebra course and am stuck on this question.

"Suppose $a$ and $b$ are non-zero vectors in $\mathbb R^n$ with the property that proj $a$ onto $b$ = proj $b$ onto $a$. What can you conclude about $a$ and $b$ $?$ Justify by describing the implied geometry and also by providing an algebraic calculation that shows your intuition is correct."

Any help would be greatly appreciated

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2 Answers

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Hint: projecting a vector onto $\bf v$ yields a vector parallel to $\bf v$.

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If we are just considering the case that the magnitudes must be the same, then a and b can be any vectors in Real space.

However, if a . b = b . a (they result in the same vector), then they have to be on the same line. If they are not, there is some angle theta by which the two-dot products are separated. Two lines of the same magnitude, starting from the same point, and separated by an angle of 0 are the same.

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