Problem
Let vector A $ = \langle a, 2\rangle$ and vector B $ = \langle 1, 3\rangle$. For what values of $a$ is the component of A along B negative?
Computationally, I know how to get the answer, but intuitively, the component of A along B being negative doesn't make sense to me. If B was along either the x-axis or y-axis, then the component of A being negative would make sense but B is not along any particular axis. B is just pointing in some random direction, so how can the length of the projection of A onto this random direction be given a sign?
$\endgroup$ 61 Answer
$\begingroup$If I understand the question with what I think is the usual, standard meaning of things, the component of $\;A\;$ along $\;B\;$ is just the length of the orthogonal projection of $\;A\;$ on the subspace $\;Span\{B\}\;$, so:
$$\text{comp}_B\,(A):=\frac{A\bullet B}{\left\|B\right\|}=\frac{(a,2)\bullet (1,3)}{\left\|(1,3)\right\|}=\frac{a+6}{\sqrt{10}}$$
End the argument now.
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