## Problem on the sign of dot products

Let $\mathbf{O}, \mathbf{N}, \mathbf{B}$ be three points as shown.
(a) What is the sign of the dot product $(\mathbf{B} - \mathbf{O}) \cdot (\mathbf{N} - \mathbf{O})$?
(b) What is the sign of the dot product $(\mathbf{B} - \mathbf{N}) \cdot (\mathbf{N} - \mathbf{O})$?
• ## Solution

#### Part (a)

Because we are interested in the dot product of the difference of vectors, we recall the geometric meaning of vector subtraction.
Now we draw the vectors $\mathbf{B} - \mathbf{O}$ and $\mathbf{N} - \mathbf{O}$.
Recall that
From the picture, the vectors appear to be separated by an angle of a little greater than $\pi/4$. Hence the dot product is positive.

#### Part (b)

We begin by drawing the vectors $\mathbf{B} - \mathbf{N}$ and $\mathbf{N} - \mathbf{O}$ .
The angle between these vectors is greater than $\pi/2$. Hence, $\cos \theta < 0$ and the dot product is negative.