Problem on the angle between vectors in 3d

Sketch all the unit vectors in 3d that have an angle of $\pi/6$ with respect to the vector $\bfi$.
• Solution

First, let's sketch all unit vectors in 3d.
Recall that
For visual simplicity, we will sketch vectors as single points, as opposed to arrows pointing from the origin.
The set of 3d points with distance 1 from the origin forms a sphere:
Now we select which of these points has angle $\pi/6$ with respect to $\bfi$.
Recall that
Because we are considering vectors as points, all our vectors share the origin as their base.
The angle of $\pi/6$ can be swept in any plane containing $\bfi$.
The set of vectors that form an angle with $\pi/6$ from $\bfi$ form a cone opening toward $\bfi$ in 3d:
The intersection of the sphere with this cone is a circle.
Hence there is a whole circle of 3d unit vectors that have angle $\pi/6$ from $\bfi$: