Components of vectors

The unit vectors i and j, components and column vector notation, magnitude and direction from components, unit vectors, operations with components, position vectors, and vectors between two points.

Components

Let $\underset{\sim}{i}$ and $\underset{\sim}{j}$ be the unit vectors in the directions of the positive $x$- and $y$-axes. Every vector in the plane can be written uniquely as $$\underset{\sim}{u} = x\underset{\sim}{i} + y\underset{\sim}{j} = \begin{pmatrix} x \\ y \end{pmatrix}.$$ The numbers $x$ and $y$ are its components. Vectors are added, subtracted and multiplied by scalars component by component.

Magnitude and direction

$$|\underset{\sim}{u}| = \sqrt{x^2 + y^2} \qquad \tan\theta = \frac{y}{x}$$

Here $\theta$ is the angle from the positive $x$-axis. Check the quadrant from the signs of $x$ and $y$. Going the other way, a vector of length $r$ at angle $\theta$ is $r(\cos\theta\,\underset{\sim}{i} + \sin\theta\,\underset{\sim}{j})$.

Example 1●

$\underset{\sim}{u} = 3\underset{\sim}{i} - 4\underset{\sim}{j}$. Find $|\underset{\sim}{u}|$ and the unit vector $\hat{u}$ in the direction of $\underset{\sim}{u}$.

$|\underset{\sim}{u}| = \sqrt{9 + 16} = 5$.

$\hat{u} = \dfrac{\underset{\sim}{u}}{|\underset{\sim}{u}|} = \tfrac35\underset{\sim}{i} - \tfrac45\underset{\sim}{j}$.

Example 2●●

Write a vector of length $10$ at $150^\circ$ to the positive $x$-axis in component form.

$10(\cos 150^\circ\,\underset{\sim}{i} + \sin 150^\circ\,\underset{\sim}{j}) = -5\sqrt3\,\underset{\sim}{i} + 5\underset{\sim}{j}$.

Position vectors

The position vector of a point $A$ is $\underset{\sim}{a} = \overrightarrow{OA}$. Then $$\overrightarrow{AB} = \underset{\sim}{b} - \underset{\sim}{a},$$ "head minus tail".

Example 3●●

$A = (1, 2)$ and $B = (4, 6)$. Find $\overrightarrow{AB}$ and the distance $AB$.

$\overrightarrow{AB} = \begin{pmatrix} 4 - 1 \\ 6 - 2 \end{pmatrix} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$.

$AB = \sqrt{3^2 + 4^2} = 5$.

Flashcards

$|xunderset{sim}{i} + yunderset{sim}{j}|$

$\sqrt{x^2 + y^2}$

Unit vector in the direction of $underset{sim}{u}$

$\hat{u} = \dfrac{\underset{\sim}{u}}{|\underset{\sim}{u}|}$

$overrightarrow{AB}$ from position vectors

$\underset{\sim}{b} - \underset{\sim}{a}$

Length $r$ at angle $ heta$

$r\cos\theta\,\underset{\sim}{i} + r\sin\theta\,\underset{\sim}{j}$

Practice

Find $|3\underset{\sim}{i} - 4\underset{\sim}{j}|$.
Find $|{-5}\underset{\sim}{i} + 12\underset{\sim}{j}|$.
$A = (1, 2)$ and $B = (4, 6)$. Find the components of $\overrightarrow{AB}$. Enter them as $x, y$.
For the same points, find the components of $\overrightarrow{BA}$.
$\underset{\sim}{u} = \begin{pmatrix} 2 \\ 1 \end{pmatrix}$ and $\underset{\sim}{v} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$. Find the components of $2\underset{\sim}{u} + \underset{\sim}{v}$. Enter them as $x, y$.
Find the $x$-component of the unit vector in the direction of $3\underset{\sim}{i} - 4\underset{\sim}{j}$.
A vector has length $10$ at $150^\circ$ to the positive $x$-axis. Find its $y$-component.
The direction of $-2\underset{\sim}{i} + 2\underset{\sim}{j}$, measured from the positive $x$-axis, is:
A unit vector has: