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# Vectors

- **vector:** a directed line segment  
- arrow indicates direction
- length indicates magnitude
- notated as $\vec{a}, \widetilde{A}, \overrightharp{a}$
- column notation: $\begin{bmatrix}
       x \\ y
     \end{bmatrix}$
- vectors with equal magnitude and direction are equivalent


![](graphics/vectors-intro.png)

## Vector addition

$\boldsymbol{u} + \boldsymbol{v}$ can be represented by drawing each vector head to tail then joining the lines.  
Addition is commutative (parallelogram)

## Scalar multiplication

For $k \in \mathbb{R}^+$, $k\boldsymbol{u}$ has the same direction as $\boldsymbol{u}$ but length is multiplied by a factor of $k$.

When multiplied by $k < 0$, direction is reversed and length is multplied by $k$.

## Vector subtraction

To find $\boldsymbol{u} - \boldsymbol{v}$, add $\boldsymbol{-v}$ to $\boldsymbol{u}$

## Parallel vectors

Parallel vectors have same direction or opposite direction.

**Two non-zero vectors $\boldsymbol{u}$ and $\boldsymbol{v}$ are parallel if there is some $k \in \mathbb{R} \setminus \{0\}$ such at $\boldsymbol{u} = k \boldsymbol{v}$**

## Position vectors

Vectors may describe a position relative to $O$.

For a point $A$, the position vector is $\overrightharp{OA}$

\vfill\eject

## Linear combinations of non-parallel vectors

If two non-zero vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are not parallel, then:

$$m\boldsymbol{a} + n\boldsymbol{b} = p \boldsymbol{a} + q \boldsymbol{b}\quad \therefore \quad m = p, \> n = q$$

![](graphics/parallelogram-vectors.jpg){#id .class width=20%}
![](graphics/vector-subtraction.jpg){#id .class width=10%}

## Column vector notation

A vector between points $A(x_1,y_1), \> B(x_2,y_2)$ can be represented as $\begin{bmatrix}x_2-x_1\\ y_2-y_1 \end{bmatrix}$

## Component notation

A vector $\boldsymbol{u} = \begin{bmatrix}x\\ y \end{bmatrix}$ can be written as $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$.  
$\boldsymbol{u}$ is the sum of two components $x\boldsymbol{i}$ and $y\boldsymbol{j}$  
Magnitude of vector $\boldsymbol{u} = x\boldsymbol{i} + y\boldsymbol{j}$ is denoted by $|u|=\sqrt{x^2+y^2}$

Basic algebra applies:  
$(x\boldsymbol{i} + y\boldsymbol{j}) + (m\boldsymbol{i} + n\boldsymbol{j}) = (x + m)\boldsymbol{i} + (y+n)\boldsymbol{j}$  
Two vectors equal if and only if their components are equal.

## Unit vectors

A vector of length 1. $\boldsymbol{i}$ and $\boldsymbol{j}$ are unit vectors.

A unit vector in direction of $\boldsymbol{a}$ is denoted by $\hat{\boldsymbol{a}}$:

$$\hat{\boldsymbol{a}}={1 \over {|\boldsymbol{a}|}}\boldsymbol{a}\quad (\implies |\hat{\boldsymbol{a}}|=1)$$

Also, unit vector of $\boldsymbol{a}$ can be defined by $\boldsymbol{a} \cdot {|\boldsymbol{a}|}$

## Scalar products / dot products

If $\boldsymbol{a} = a_i \boldsymbol{i} + a_2 \boldsymbol{j}$ and $\boldsymbol{b} = b_i \boldsymbol{i} + b_2 \boldsymbol{j}$, the dot product is:
$$\boldsymbol{a} \cdot \boldsymbol{b} = a_1 b_1 + a_2 b_2$$

Produces a real number, not a vector.

$$\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$$

**on CAS:** `dotP([a b c], [d e f])`

## Scalar product properties

1. $k(\boldsymbol{a\cdot b})=(k\boldsymbol{a})\cdot \boldsymbol{b}=\boldsymbol{a}\cdot (k{b})$
2. $\boldsymbol{a \cdot 0}=0$
3. $\boldsymbol{a \cdot (b + c)}=\boldsymbol{a \cdot b + a \cdot c}$
4. $\boldsymbol{i \cdot i} = \boldsymbol{j \cdot j} = \boldsymbol{k \cdot k}= 1$
5. If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular
6. $\boldsymbol{a \cdot a} = |\boldsymbol{a}|^2 = a^2$

For parallel vectors $\boldsymbol{a}$ and $\boldsymbol{b}$:  
$$\boldsymbol{a \cdot b}=\begin{cases}
|\boldsymbol{a}||\boldsymbol{b}| \hspace{2.8em} \text{if same direction}\\
-|\boldsymbol{a}||\boldsymbol{b}| \hspace{2em} \text{if opposite directions}
\end{cases}$$

## Geometric scalar products

$$\boldsymbol{a} \cdot \boldsymbol{b} = |\boldsymbol{a}| |\boldsymbol{b}| \cos \theta$$

where $0 \le \theta \le \pi$

## Perpendicular vectors

If $\boldsymbol{a} \cdot \boldsymbol{b} = 0$, then $\boldsymbol{a} \perp \boldsymbol{b}$ (since $\cos 90 = 0$)

## Finding angle between vectors

**positive direction**

$$\cos \theta = {{\boldsymbol{a} \cdot \boldsymbol{b}} \over {|\boldsymbol{a}| |\boldsymbol{b}|}} = {{a_1 b_1 + a_2 b_2} \over {|\boldsymbol{a}| |\boldsymbol{b}|}}$$

**on CAS:** `angle([a b c], [a b c])` (Action -> Vector -> Angle)


## Vector projections

Vector resolute of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$ is magnitude of $\boldsymbol{a}$ in direction of $\boldsymbol{b}$.

$$\boldsymbol{u}={{\boldsymbol{a}\cdot\boldsymbol{b}}\over |\boldsymbol{b}|^2}\boldsymbol{b}=\left({\boldsymbol{a}\cdot{\boldsymbol{b} \over |\boldsymbol{b}|}}\right)\left({\boldsymbol{b} \over |\boldsymbol{b}|}\right)=(\boldsymbol{a} \cdot \hat{\boldsymbol{b}})\hat{\boldsymbol{b}}$$

Scalar resolute of $\vec{a}$ on $\vec{b} = |\vec{u}| = \vec{a} \cdot \hat{\vec{b}}$ (results in a scalar)  
Vector resolute of $\vec{a}$ perpendicular to $b$ is equal to $\vec{a} - \vec{u}$ where $\vec{u}$ is vector projection of $\vec{a}$ on $\vec{b}$

## Vector proofs

**Concurrent lines -** $\ge$ 3 lines intersect at a single point  
**Collinear points -** $\ge$ 3 points lie on the same line ($\implies \vec{OC} = \lambda \vec{OA} + \mu \vec{OB}$ where $\lambda + \mu = 1$. If $C$ is between $\vec{AB}$, then $0 \lt \mu \lt 1$)

Useful vector properties:

- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel, then $\boldsymbol{b}=k\boldsymbol{a}$ for some $k \in \mathbb{R} \setminus \{0\}$
- If $\boldsymbol{a}$ and $\boldsymbol{b}$ are parallel with at least one point in common, then they lie on the same straight line
- Two vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ are perpendicular if $\boldsymbol{a} \cdot \boldsymbol{b}=0$
- $\boldsymbol{a} \cdot \boldsymbol{a} = |\boldsymbol{a}|^2$

## Linear dependence

Vectors $\vec{a}, \vec{b}, \vec{c}$ are linearly dependent if they are non-parallel and:

$$k\vec{a}+l\vec{b}+m\vec{c} = 0$$
$$\therefore \vec{c} = m\vec{a} + n\vec{b} \quad \text{(simultaneous)}$$

$\vec{a}, \vec{b},$ and $\vec{c}$ are linearly independent if no vector in the set is expressible as a linear combination of other vectors in set, or if they are parallel.

Vector $\vec{w}$ is a linear combination of vectors $\vec{v_1}, \vec{v_2}, \vec{v_3}$

## Three-dimensional vectors

Right-hand rule for axes - $z$ is up or out of page.

## Angle between vector and axis

Direction of a vector can be given by the angles it makes with $\vec{i}, \vec{j}, \vec{k}$ directions.

For $\vec{a} = a_1 \vec{i} + a_2 \vec{j} + a_3 \vec{k}$ which makes angles $\alpha, \beta, \gamma$ with positive direction of $x, y, z$ axes:
$$\cos \alpha = {a_1 \over |\vec{a}|}, \quad \cos \beta = {a_2 \over |\vec{a}|}, \quad \cos \gamma = {a_3 \over |\vec{a}|}$$

**on CAS:** `angle([a b c], [1 0 0])` for angle between $a\vec{i} + b\vec{j} + c\vec{k}$ and $x$-axis

## Collinearity

Points $A, B, C$ are collinear iff $\vec{AC}=m\vec{AB} \text{ where } m \ne 0$