## Problem on the sign of the component along a vector

Show that the component (scalar projection) of $\bfa$ along $\bfb$ is positive if the angle between $\bfa$ and $\bfb$ is less than $\pi/2$. Show that it is negative if the angle is greater than $\pi/2$.
• ## Solution

Recall that
Hence the sign of the component of $\bfa$ along $\bfb$ equals the sign of the dot product of $\bfa$ and $\bfb$.
Recall the relationship of dot product and angle:
We see that the sign of $\bfa \cdot \bfb$ is the same as the sign of $\cos \theta$.
We conclude that $\text{comp}_\bfb \bfa$ is positive when $\theta < \pi/2$ and negative when $\theta > \pi/2$.