## Problem on dot products

Of the unit vectors $\mathbf{A}$ through $\mathbf{H}$,
1. Which have a positive dot product with $\mathbf{A}$?

2. Which have a negative dot product with $\mathbf{A}$?

3. Which have zero dot product with $\mathbf{A}$?

4. Which has the largest dot product with $\mathbf{A}$?

5. Which has the most negative dot product with $\mathbf{A}$?

• ## Solution

We are given that the vectors have unit length. The only other information we have is a visual estimate that their angles are spaced every $\pi/4$ radians. To make use of this information, we need to relate dot product to length and angle.
Recall the dot product angle formula:
(b) We see from (1) that a dot product is negative when $\pi/2 < \theta \leq \pi$. The vectors that are within this range of angles are: $$\mathbf{D}, \mathbf{E}, \mathbf{F}$$
(d) From (1) and the fact that all of the vectors have unit length, the one with the largest dot product will be the one with the smallest angle from $\mathbf{A}$. The vector $\mathbf{A}$ has angle $\theta=0$ relative to $\mathbf{A}$, so it is the vector with the largest dot product with $\mathbf{A}$.
(e) From (1) and the fact that all the vectors have unit length, the one with the most negative dot product will be the one with the largest angle from $\mathbf{A}$. The largest possible value of $\theta$ is $\pi$, which is achieved by $\mathbf{E}$.