Study guide and
7 practice problems
on:
Angle between vectors
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Study Guide
The angle between two vectors is the angle swept by the arc that directly connects them, provided that the vectors share the same base.
(7 problems)
The angle between vectors is always between 0 and $\pi$, inclusive. It is 0 if the vectors are in the same direction and $\pi$ if the vectors are in opposite directions.
(2 problems)
Related topics
Vectors
(55 problems)
Multivariable calculus
(147 problems)
Practice problems
Sketch all 3d vectors whose angle with respect to the vector $\bfi$ is
$\pi/6$
$\pi/2$
$5\pi/6$
Solution
Sketch all the unit vectors in 3d that have an angle of $\pi/6$ with respect to the vector $\bfi$.
Solution
Estimate the angle between the following pairs of vectors:
Solution
Sketch all the unit vectors in 2d that have an angle of $\pi/4$ with respect to the vector $\bfi$.
Solution
Find the angle at the apex of a triangular faces of the pyramid formed by the points $(1, 1, 0)$, $(1,-1, 0)$, $(-1, 1, 0)$, $(-1, -1, 0)$, and $(0,0,1)$.
Solution
Prove that $\bfx \cdot \bfx = \left| \bfx \right|^2$ in two ways:
Directly (in the case of 3d vectors)
By the dot product angle formula
Solution
What is the angle between a nonzero vector $\bfx$ and $-\bfx$? Use that angle to show that $\bfx \times (-\bfx)=0$.
Solution