Study guide and
8 practice problems
on:
Dot product angle formula
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If $\theta$ is the angle between two vectors $\bfx$ and $\bfy$, $$\mathbf{x} \cdot \mathbf{y} = \left | \mathbf{x} \right | \left | \mathbf{y} \right | \cos \theta$$
Related topics
Dot product
(41 problems)
Multivariable calculus
(147 problems)
Practice problems
Prove that $\cos (\theta_1 + \theta_2) = \cos \theta_1 \cos \theta_2 - \sin \theta_1 \sin \theta_2$ by considering the dot product of the unit vectors $\mathbf{v_1}$ and $\mathbf{v_2}$. These vectors are at the angles $\theta_1$ above the $x$-axis and $\theta_2$ below the $x$-axis, respectively.
Solution
Let $\mathbf{O}, \mathbf{N}, \mathbf{B}$ be three points as shown.
(a) What is the sign of the dot product $(\mathbf{B} - \mathbf{O}) \cdot (\mathbf{N} - \mathbf{O})$?
(b) What is the sign of the dot product $(\mathbf{B} - \mathbf{N}) \cdot (\mathbf{N} - \mathbf{O})$?
Solution
Of the unit vectors $\mathbf{A}$ through $\mathbf{H}$,
Which have a positive dot product with $\mathbf{A}$?
Which have a negative dot product with $\mathbf{A}$?
Which have zero dot product with $\mathbf{A}$?
Which has the largest dot product with $\mathbf{A}$?
Which has the most negative dot product with $\mathbf{A}$?
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
Derive the law of cosines using the dot product:
(a) Write $\text{CB}$ in terms of $\text{OB}$ and $\text{OC}$
(b) Write $\left | \text{CB} \right|^2$ in terms of $\left | \text{OB} \right |$, $\left | \text{OC} \right |$ and $\text{OB} \cdot \text{OC}$
(c) Show that $\left | \text{CB} \right|^2 = \left | \text{OB} \right|^2 + \left | \text{OC} \right |^2 - 2 \left | \text{OB} \right| \left | \text{OC} \right| \cos \theta$.
Solution
Compute the vector projection of $\bfi$ onto $\bfi + \bfj$.
Solution
Show that the component (scalar projection) of $\bfa$ along $\bfb$ is positive if the angle between $\bfa$ and $\bfb$ is less than $\pi/2$. Show that it is negative if the angle is greater than $\pi/2$.
Solution