## Problem on the angle between vectors in 3d

Sketch all 3d vectors whose angle with respect to the vector $\bfi$ is
1. $\pi/6$
2. $\pi/2$
3. $5\pi/6$
• ## Solution

Recall that
We consider vectors that share the origin as their base.

#### Part (a)

In any plane containing $\bfk$, the vectors that have an angle $\pi/6$ with respect to $\bfk$ are:
There are many planes containing $\bfk$, such as:
Putting together the vectors over all possible planes containing $\bfk$, we get that the vectors with angle $\pi/6$ from $\bfk$ form a cone:

#### Part (b)

In any plane containing $\bfk$, the vectors that have an angle $\pi/2$ with respect to $\bfk$ are:
Considering all possible planes containing $\bfk$, the vectors with angle $\pi/2$ from $\bfk$ are a plane:

#### Part (c)

In any plane containing $\bfk$, the vectors that have an angle $5\pi /6$ with respect to $\bfk$ are:
Considering all possible planes containing $\bfk$, the vectors with angle $5\pi/6$ from $\bfk$ are a cone that opens away from $\bfk$: