Study guide and
7 practice problems
on:
Definition of the angle between two vectors
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The angle between two vectors is the angle swept by the arc that directly connects them, provided that the vectors share the same base.
In three dimensions, two vectors define a plane, and the arc connecting them lives in that plane.
Related topics
Angle between vectors
(7 problems)
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