Problem on the angle between vectors in 2d
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Sketch all the unit vectors in 2d that have an angle of $\pi/4$ with respect to the vector $\bfi$.
Solution
First, let's sketch all unit vectors in 2d.
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, instead of arrows pointing from the origin.
The set of 2d points of distance 1 from the origin forms a circle:
Now we select which of these points has angle $\pi/4$ 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 that 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 $\pi/4$ can be swept in either the clockwise or counterclockwise directions.
We sketch all vectors of angle $\pi/4$ from $\bfi$:
Hence, there are two 2d unit vectors with angle $\pi/4$ from $\bfi$:
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