Problem on the angle between vectors in 3d
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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
Definition of a unit vector
A unit vector is a vector of length 1.
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
Definition of the angle between two vectors
The angle between two vectors is the angle swept by the arc that directly connects them, provided the vectors share the same base.
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 $\bfi$:
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