Study guide and
6 practice problems
on:
Finding a perpendicular vector
$\newcommand{\bfA}{\mathbf{A}}$ $\newcommand{\bfB}{\mathbf{B}}$ $\newcommand{\bfC}{\mathbf{C}}$ $\newcommand{\bfF}{\mathbf{F}}$ $\newcommand{\bfI}{\mathbf{I}}$ $\newcommand{\bfa}{\mathbf{a}}$ $\newcommand{\bfb}{\mathbf{b}}$ $\newcommand{\bfc}{\mathbf{c}}$ $\newcommand{\bfd}{\mathbf{d}}$ $\newcommand{\bfe}{\mathbf{e}}$ $\newcommand{\bfi}{\mathbf{i}}$ $\newcommand{\bfj}{\mathbf{j}}$ $\newcommand{\bfk}{\mathbf{k}}$ $\newcommand{\bfn}{\mathbf{n}}$ $\newcommand{\bfr}{\mathbf{r}}$ $\newcommand{\bfu}{\mathbf{u}}$ $\newcommand{\bfv}{\mathbf{v}}$ $\newcommand{\bfw}{\mathbf{w}}$ $\newcommand{\bfx}{\mathbf{x}}$ $\newcommand{\bfy}{\mathbf{y}}$ $\newcommand{\bfz}{\mathbf{z}}$
There are multiple ways to find a vector perpendicular to the vector $\mathbf{x}$:
A vector perpendicular to $\mathbf{x} = \langle a,b \rangle$ is $\langle -b, a \rangle$.
Write and solve the equation $\mathbf{x} \cdot \mathbf{y} = 0$.
Take the cross product with any vector that is not a multiple of $\mathbf{x}$.
Related topics
Vectors
(55 problems)
Multivariable calculus
(147 problems)
Practice problems
Find a 2d vector that is perpendicular to $\langle 2,3 \rangle$. Verify that it is perpendicular.
Solution
Consider the point $\bfx_0 = (x_0, y_0)$ and the line given by $\bfn \cdot \bfx = 0$, where $\bfn = (a, b)$. Using a vector projection, find the coordinates of the nearest point to $\bfx_0$ on the line $\bfn\cdot \bfx =0$.
Solution
Find a vector perpendicular to $\langle1, 2, 3 \rangle$.
Solution
Consider a cylinder of radius $r$ rolling up a hill of incline $\theta$ at constant speed $v$. Initially the point of contact is $(0,0)$. Find the trajectory of the point initially contacting the hill.
Solution
Find the line containing the points $(0, 1)$ and $(3, 0)$ by computing a normal vector to the line. Write it in the form $\bfn \cdot \bfx = b$.
Solution
Let $f(x,y) = x^2 + y^3 $ and $P=(1,1)$.
Find a direction at $P$ along which $f$ is not changing.
Solution