Category: Lessons (Page 3 of 4)

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

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

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 $\overrightarrow{v}$ (it is written by some authors as $v$ ), its magnitude is denoted by $||\overrightarrow{v}||,$ and its directional angle by $\theta$.

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

We can always represent a vector $\overrightarrow{v}$ by arranging the starting point of $\overrightarrow{v}$ to be the origin $O(0,0)$ (as in $\overrightarrow{OR}$ in the picture above). If $R$ has coordinates $R(a,b)$ then we also write for $\overrightarrow{v}=\overrightarrow{OR}$:
$\overrightarrow{v}=⟨a,b⟩$, or $\overrightarrow{v}=\left[\begin{array}{c}a\\ b\end{array}\right]$

Example 22.2. Graph the vectors $\overrightarrow{v},\overrightarrow{w},\overrightarrow{r},\overrightarrow{s},\overrightarrow{t}$ in the plane, where
$\overrightarrow{v}=\overrightarrow{PQ}$ with $P(6,3)$ and $Q(4,-2),$ and
$\overrightarrow{w}=⟨3,-1⟩,\overrightarrow{r}=⟨-4,-2⟩,\overrightarrow{s}=⟨0,2⟩,\overrightarrow{t}=⟨-5,3⟩$

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:

Example 22.4. Find the magnitude and directional angle of the given vectors:
a) $⟨-6,6⟩$
b) $⟨4,-3⟩$
c) $⟨-2\sqrt{3},-2⟩$
d) $⟨8,4\sqrt{5}⟩$
e) $\overrightarrow{PQ}$, where $P(9,2)$ and $Q(3,10)$

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 $\overrightarrow{v}=⟨a,b⟩$ is defined to be the vector given by multiplying each coordinate by $r$:
$r⟨a,b⟩=⟨r\cdot a,r\cdot b⟩$

Example 22.6. Multiply, and graph the vectors
a) $4\cdot ⟨-2,1⟩$
b) $(-3)\cdot ⟨-6,-2⟩$

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

The Unit Vector

Definition 22.8. A vector $\overrightarrow{u}$ is called a unit vector if it has a magnitude of 1
$\overrightarrow{u}$ is a unit vector $⟺||\overrightarrow{u}||=1$

There are two special unit vectors $\overrightarrow{i}$ and $\overrightarrow{j}$, which are the vectors pointing in the $x-$ and the $y$ -direction.
$\overrightarrow{i}=⟨1,0⟩\text{ and }\overrightarrow{j}=⟨0,1⟩$

Example 22.9. Find a unit vector in the direction of $\overrightarrow{v}$
a) $⟨8,6⟩$
b) $⟨-2,3\sqrt{7}⟩$

Vector Addition

The second operation on vectors is called vector addition.

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

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

Example 22.11. Perform the vector addition and simplify as much as possible.
a) $⟨3,-5⟩+⟨6,4⟩$
b) $5\cdot ⟨-6,2⟩-7\cdot ⟨1,-3⟩$
c) $4\overrightarrow{i}+9\overrightarrow{j}$
d) find $2\overrightarrow{v}+3\overrightarrow{w}$ for $\overrightarrow{v}=-6\overrightarrow{i}-4\overrightarrow{j}$ and $\overrightarrow{w}=10\overrightarrow{i}-7\overrightarrow{j}$
e) find $-3\overrightarrow{v}+5\overrightarrow{w}$ for $\overrightarrow{v}=⟨8,\sqrt{3}⟩$ and $\overrightarrow{w}=⟨0,4\sqrt{3}⟩$

Example 22.12. The forces ${\overrightarrow{F}}_1$ and ${\overrightarrow{F}}_2$ are applied to an object. Find the resulting total force $\overrightarrow{F}={\overrightarrow{F}}_1+{\overrightarrow{F}}_2$. Determine the magnitude and directional angle of the total force $\overrightarrow{F}$. Approximate these values as necessary. Recall that the international system of units for force is the newton $[1N=1\frac{kg\cdot m}{s^2}]$
a) ${\overrightarrow{F}}_1$ has magnitude 3 newtons, and angle ${\theta }_1={45}^{\circ }$
${\overrightarrow{F}}_2$ has magnitude 5 newtons, and angle ${\theta }_2={135}^{\circ }$
b) $\left|{\overrightarrow{F}}_1\right|=7$ newtons, and ${\theta }_1=\frac{\pi }{6},$ and $\left|{\overrightarrow{F}}_2\right|=4$ newtons, and ${\theta }_2=\frac{5\pi }{3}$

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