Hi everyone! Read through the material below, watch the videos.

Lesson 22: Vectors in the plane

Topic: This lesson covers Chapter 22: Vectors in the plane.

WeBWorK: There are four WeBWorK assignments on today’s material:Vectors – Components

Vectors – Operations

Vectors – Magnitude and Direction

Vectors – Unit Vectors

#### Lesson Notes (Notability – pdf):

This .pdf file contains most of the work from the videos in this lesson. It is provided for your reference.

## Introduction to Vectors

Today we will be working with the plane $\mathbb{R}^2$, but looking at things in a slightly different way – instead of *points* (which have only a location), we will be focussing on *vectors* (which have a magnitude (size) and direction). This change in perspective is quite powerful, and brings to light many useful features of the plane – but in practice, you will find it similar to the work we did in the previous lesson on polar form of complex numbers.

**Definition 22.1**. A **geometric vector in the plane** is a geometric object in the plane $\mathbb{R}^{2}$ that is given by a direction (angle) and magnitude (size). We denote a vector by $\vec{v}$ (it is written by some authors as $v$ ), its magnitude is denoted by $||\vec{v}||,$ and its directional angle by $\theta$.

Vectors are often drawn as directed line segments $\vec{v}=\overrightarrow{P Q}$. Two such segments represent the same vector if they have the same magnitude and direction.

We can always represent a vector $\vec{v}$ by arranging the starting point of $\vec{v}$ to be the origin $O(0,0)$ (as in $\overrightarrow{O R}$ in the picture above). If $R$ has coordinates $R(a, b)$ then we also write for $\vec{v}=\overrightarrow{O R}$:

$\vec{v}=\langle a, b\rangle$, or $\vec{v}=\begin{bmatrix} a\\ b\end{bmatrix}$

**Example 22.2**. Graph the vectors $\vec{v}, \vec{w}, \vec{r}, \vec{s}, \vec{t}$ in the plane, where

$\vec{v}=\overrightarrow{P Q}$ with $P(6,3)$ and $Q(4,-2),$ and

$\vec{w}=\langle 3,-1\rangle, \quad \vec{r}=\langle-4,-2\rangle, \quad \vec{s}=\langle 0,2\rangle, \quad \vec{t}=\langle-5,3\rangle$

The formulas for magnitude and directional angle of a vector are the same as those for modulus (magnitude) and argument (angle) of a complex number:

**Formulas for magnitude and angle of a vector:** Suppose $\vec{v}=\langle{a,b}\rangle}$ is a vector in the plane $\mathbb{R}^2$. Then the magnitude and angle of $\vec{v}$ are given by:

$||\vec{v}||=\sqrt{a^2+b^2}$, $\tan(\theta)=\frac{b}{a}$

Conversely, we can obtain the coordinates of a vector from its magnitude and directional angle by:

$\vec{v}=\langle||\vec{v}|| \cdot \cos (\theta),||\vec{v}|| \cdot \sin (\theta)\rangle$

**Example 22.4**. Find the magnitude and directional angle of the given vectors:

a) $\langle-6,6\rangle$

b) $\langle 4,-3\rangle$

c) $\langle-2 \sqrt{3},-2\rangle$

d) $\langle 8,4 \sqrt{5}\rangle$

e) $\overrightarrow{P Q}$, where $P(9,2)$ and $Q(3,10)$

*VIDEO: Intro to vectors, finding magnitude and direction – Example 22.4*

## Operations on Vectors

There are two basic operations on vectors, *scalar multiplication* and *vector addition*.

#### Scalar Multiplication

**Definition 22.5**. The scalar multiplication of a real number $r$ with a vector $\vec{v}=\langle a, b\rangle$ is defined to be the vector given by multiplying each coordinate by $r$:

$r \langle a, b\rangle =\langle r \cdot a, r \cdot b\rangle$

**Example 22.6**. Multiply, and graph the vectors

a) $4 \cdot\langle-2,1\rangle$

b) $(-3)\cdot\langle-6,-2\rangle$

*VIDEO: Scalar multiplication of vectors – Example 22.6*

**Observation**. When we multiply a vector $\vec{v}$ by a positive real number $r>0$, the result will have the same *angle* as $\vec{v}$, while the *magnitude* will be stretched by a factor of $r$.

#### The Unit Vector

**Definition 22.8**. A vector $\vec{u}$ is called a **unit vector** if it has a magnitude of 1

$\vec{u}$ is a unit vector $\quad \Longleftrightarrow ||\vec{u}||=1$

There are two special unit vectors $\vec{i}$ and $\vec{j}$, which are the vectors pointing in the $x-$ and the $y$ -direction.

$\vec{i}=\langle 1,0\rangle \quad \text { and } \quad \vec{j}=\langle 0,1\rangle$

**Example 22.9**. Find a unit vector in the direction of $\vec{v}$

a) $\langle 8,6\rangle$

b) $\langle-2,3 \sqrt{7}\rangle$

*VIDEO: Unit vectors – Example 22.9*

#### Vector Addition

The second operation on vectors is called vector addition.

**Definition 22.10**. Let $\vec{v}=\langle a, b\rangle$ and $\vec{w}=\langle c, d\rangle$ be two vectors. Then the vector addition $\vec{v}+\vec{w}$ is defined by component-wise addition:

$\langle a, b\rangle+\langle c, d\rangle:=\langle a+c, b+d\rangle$

In the plane, this corresponds to starting at the origin, following $\vec{v}$ and then $\vec{w}$ (or vice versa, following $\vec{w}$ and then $\vec{v}$). In the picture, note that whichever path you take from the origin you will still arrive at the same point in the upper right, $\vec{v}+\vec{w}$:

**Example 22.11**. Perform the vector addition and simplify as much as possible.

a) $\langle 3,-5\rangle+\langle 6,4\rangle$

b) $5 \cdot\langle-6,2\rangle-7 \cdot\langle 1,-3\rangle$

c) $4 \vec{i}+9 \vec{j}$

d) find $2 \vec{v}+3 \vec{w}$ for $\vec{v}=-6 \vec{i}-4 \vec{j}$ and $\vec{w}=10 \vec{i}-7 \vec{j}$

e) find $-3 \vec{v}+5 \vec{w}$ for $\vec{v}=\langle 8, \sqrt{3}\rangle$ and $\vec{w}=\langle 0,4 \sqrt{3}\rangle$

*VIDEO: Vector addition – Example 22.11*

**Example 22.12**. The forces $\vec{F}_{1}$ and $\vec{F}_{2}$ are applied to an object. Find the resulting total force $\vec{F}=\vec{F}_{1}+\vec{F}_{2}$. Determine the magnitude and directional angle of the total force $\vec{F}$. Approximate these values as necessary. Recall that the international system of units for force is the newton $\left[1 N=1 \frac{k g \cdot m}{s^{2}}\right]$

a) $\vec{F}_{1}$ has magnitude 3 newtons, and angle $\theta_{1}=45^{\circ}$

$\vec{F}_{2}$ has magnitude 5 newtons, and angle $\theta_{2}=135^{\circ}$

b) $\left|\vec{F}_{1}\right|=7$ newtons, and $\theta_{1}=\frac{\pi}{6},$ and $\left|\vec{F}_{2}\right|=4$ newtons, and $\theta_{2}=\frac{5 \pi}{3}$

*VIDEO: Vector addition application (force) – Example 22.12*

That’s it for now – give the WeBWorK a try!

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