Problem on the angle between vectors in 3d
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Sketch all 3d vectors whose angle with respect to the vector $\bfi$ is
$\pi/6$
$\pi/2$
$5\pi/6$
Solution
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.
In three dimensions, two vectors with the same base define a plane, and the arc connecting them lives in that plane.
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 $\bfk$:
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