Study guide and
14 practice problems
on:
Subtraction gives the vector between two points
$\newcommand{\bfA}{\mathbf{A}}$ $\newcommand{\bfB}{\mathbf{B}}$ $\newcommand{\bfC}{\mathbf{C}}$ $\newcommand{\bfF}{\mathbf{F}}$ $\newcommand{\bfI}{\mathbf{I}}$ $\newcommand{\bfa}{\mathbf{a}}$ $\newcommand{\bfb}{\mathbf{b}}$ $\newcommand{\bfc}{\mathbf{c}}$ $\newcommand{\bfd}{\mathbf{d}}$ $\newcommand{\bfe}{\mathbf{e}}$ $\newcommand{\bfi}{\mathbf{i}}$ $\newcommand{\bfj}{\mathbf{j}}$ $\newcommand{\bfk}{\mathbf{k}}$ $\newcommand{\bfn}{\mathbf{n}}$ $\newcommand{\bfr}{\mathbf{r}}$ $\newcommand{\bfu}{\mathbf{u}}$ $\newcommand{\bfv}{\mathbf{v}}$ $\newcommand{\bfw}{\mathbf{w}}$ $\newcommand{\bfx}{\mathbf{x}}$ $\newcommand{\bfy}{\mathbf{y}}$ $\newcommand{\bfz}{\mathbf{z}}$
The vector from $\bfx$ to $\bfy$ is given by $\bfy - \bfx$.
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Practice problems
Consider a pyramid with square base formed by the points $(1,1,0), (1,-1, 0), (-1, 1, 0), (-1, -1, 0),$ and $(0,0,1)$. What is the length of each edge connecting the base to the apex?
Solution
Find the length of the vector from $(2,4,5)$ to $(3, -1, -2)$.
Solution
Let $\mathbf{z}$ be the point one third of the way from $\mathbf{x}$ to $\mathbf{y}$. Using vector arithmetic, express $\mathbf{z}$ in terms of $\mathbf{x}$ and $\mathbf{y}$.
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
Use vector addition, subtraction, and scalar multiplication to show that the midpoint between the two points $\mathbf{x}$ and $\mathbf{y}$ is $\frac{\mathbf{x} + \mathbf{y}}{2}$.
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
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
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
Show that the lines connecting any point on the semicircle of radius 1 to $(1,0)$ and $(-1,0)$ are perpendicular.
Solution
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