Study guide and
41 practice problems
on:
Dot product
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Study Guide
Dot product is defined by $\langle x_1, x_2, x_3 \rangle \cdot \langle y_1, y_2, y_3 \rangle = x_1 y_1 + x_2 y_2 + x_3 y_3.$
(21 problems)
$\mathbf{x} \cdot \mathbf{y} = \left | \mathbf{x} \right | \left | \mathbf{y} \right | \cos \theta$.
(8 problems)
The vectors $\mathbf{x}$ and $\mathbf{y}$ are perpendicular if and only if $\mathbf{x} \cdot \mathbf{y} = 0$.
(13 problems)
Dot product of vectors is positive if they point in the same 'general direction.'
(6 problems)
$\mathbf{x} \cdot \mathbf{x} = |\mathbf{x}|^2.$
(7 problems)
Algebra of dot products
$$\begin{align}
\mathbf{x} \cdot ( \mathbf{y} + \mathbf{z}) &= \mathbf{x} \cdot \mathbf{y} + \mathbf{x} \cdot \mathbf{z}\\
(\mathbf{w} + \mathbf{x})\cdot(\mathbf{y} + \mathbf{z}) &= \mathbf{w}\cdot \mathbf{y} + \mathbf{w}\cdot \mathbf{z} + \mathbf{x} \cdot \mathbf{y} + \mathbf{x} \cdot \mathbf{z}\\
\mathbf{x}\cdot \mathbf{y} &= \mathbf{y}\cdot \mathbf{x}
\end{align}$$
(6 problems)
Geometric proofs with vectors
(6 problems)
Begin a geometric proof by labeling important points with as few variables as possible.
(5 problems)
The midpoint between the two vectors $\mathbf{x}$ and $\mathbf{y}$ is $\frac{\mathbf{x} + \mathbf{y}}{2}$.
(3 problems)
The component of $\bfx$ along $\bfv$ is $$\text{comp}_\bfv \bfx = \frac{\bfx\cdot \bfv}{\left| \bfv \right|}.$$
(4 problems)
The vector projection of $\bfx$ onto $\bfv$ is $$\text{proj}_{\bfv}\ \bfx = (\bfx\cdot \bfv) \frac{\bfv}{\left| \bfv \right|^2}.$$
(4 problems)
Related topics
Multivariable calculus
(147 problems)
Practice problems
Find a 2d vector that is perpendicular to $\langle 2,3 \rangle$. Verify that it is perpendicular.
Solution
Suppose that two opposite sides of a quadrilateral are parallel and have equal length. Show that the quadrilateral is a parallelogram.
Solution
Consider an arbitrary quadrilateral. The two blue line segments connect the midpoints of adjacent sides. Using only vector addition and multiplication by constants, show that these line segments are parallel and have the same length.
Solution
Show that the line connecting the midpoints of two sides of a triangle is parallel to and half the length of the third side.
Solution
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
If $\bfx \cdot \bfy = \bfx \cdot \bfz$ for a nonzero $\bfx$, does $\bfy = \bfz$? If so, prove it. If not, provide a counterexample.
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
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