Study guide and
6 practice problems
on:
Vector-scalar multiplication scales the length of the vector
$\newcommand{\bfA}{\mathbf{A}}$ $\newcommand{\bfB}{\mathbf{B}}$ $\newcommand{\bfC}{\mathbf{C}}$ $\newcommand{\bfF}{\mathbf{F}}$ $\newcommand{\bfI}{\mathbf{I}}$ $\newcommand{\bfa}{\mathbf{a}}$ $\newcommand{\bfb}{\mathbf{b}}$ $\newcommand{\bfc}{\mathbf{c}}$ $\newcommand{\bfd}{\mathbf{d}}$ $\newcommand{\bfe}{\mathbf{e}}$ $\newcommand{\bfi}{\mathbf{i}}$ $\newcommand{\bfj}{\mathbf{j}}$ $\newcommand{\bfk}{\mathbf{k}}$ $\newcommand{\bfn}{\mathbf{n}}$ $\newcommand{\bfr}{\mathbf{r}}$ $\newcommand{\bfu}{\mathbf{u}}$ $\newcommand{\bfv}{\mathbf{v}}$ $\newcommand{\bfw}{\mathbf{w}}$ $\newcommand{\bfx}{\mathbf{x}}$ $\newcommand{\bfy}{\mathbf{y}}$ $\newcommand{\bfz}{\mathbf{z}}$
$c \mathbf{x}$ is the vector obtained by multiplying the length of $\bfx$ by $c$.
If $c < 0$, then the direction of the vector becomes reversed.
For example, if $c = \pm1.5$:
Related topics
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Practice problems
Estimate what positive multiples of the following vectors add up to zero.
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
For a scalar $c$ and a vector $\bfx$, show that $\left |c\bfx \right| = \left |c\right | \ \left |\bfx \right|.$
Solution
Suppose that two opposite sides of a quadrilateral are parallel and have equal length. Show that the quadrilateral is a parallelogram.
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
Consider the triangle in three-space given by $\langle a, 0, 0\rangle, \langle 0, b, 0 \rangle, \langle 0, 0, c \rangle.$ Find a vector that is perpendicular to the triangle and has length equal to the area of the triangle.
Solution