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:
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
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