Introduction to Vectors

In short: A vector has a size (magnitude) and a direction, and is written as $A B$ , $a$ or $x i + y j$ . Vectors add and subtract component by component, and the magnitude of $x i + y j$ is $x^{2} + y^{2}$ .

Starting point: what you already know

Pythagoras and SOH CAH TOA.
Coordinates: the point $(3, 4)$ is 3 right and 4 up from the origin.
True bearings and the difference between distance (how far) and displacement (how far and which way), from science.
Gradient: lines with the same gradient are parallel.

Scalars and vectors

A scalar has size only: mass 5 kg, time 10 s, temperature $2 0^{\circ} C$ .

A vector has size and direction: a displacement of 5 km north, a velocity of 30 m/s east, a force of 10 N downward.

We draw a vector as an arrow. Its length shows the size, and the arrowhead shows the direction.

Writing vectors

As a directed line segment from $A$ to $B$ : $A B$ .
As a bold letter: $a$ . By hand, write $\sim a$ with a squiggle under it.
In component form: $a = (34)$ or $a = 3 i + 4 j$ .

Here $i = (10)$ is one unit right and $j = (01)$ is one unit up.

For points $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ : $A B = (x_{2} - x_{1} y_{2} - y_{1})$ "Where you finish minus where you start."

Magnitude and direction

The magnitude (length) of $a = x i + y j$ is

$∣ a ∣ = x^{2} + y^{2}$

and its direction angle $θ$ from the positive x-axis satisfies $tan θ = \frac{y}{x}$ . Check the quadrant from a sketch.

Adding and subtracting

Add vectors head to tail: start the second where the first ends. In components, just add.

$(a_{1} a_{2}) + (b_{1} b_{2}) = (a_{1} + b_{1} a_{2} + b_{2})$

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Subtract: $a - b = a + (- b)$ , where $- b$ has the same length as $b$ but points the opposite way.

Scalar multiples and unit vectors

$k a$ stretches $a$ by a factor $k$ . If $k < 0$ it also flips the direction. Two vectors are parallel when one is a scalar multiple of the other.

A unit vector has length 1. The unit vector in the direction of $a$ is

$a^= \frac{a}{∣ a ∣}$

Vectors

Meaning

Size and direction
Scalars have size only

Notation

AB with an arrow
Bold a
Column or i, j form

Size

|a| = √(x² + y²)
Direction tan θ = y/x

Operations

Add head to tail
Subtract adds the opposite
ka scales

Special

Unit vector a/|a|
Parallel means a = kb
Position vectors from O

Year 12Position vectors and midpoints

The position vector of a point $P$ is $O P$ , from the origin. If $A$ and $B$ have position vectors $a$ and $b$ , then $A B = b - a, midpoint M : O M = \frac{1}{2} (a + b)$ Example. Show $A (1, 2)$ , $B (3, 5)$ and $C (7, 11)$ lie on one line. $A B = 2 i + 3 j$ and $A C = 6 i + 9 j = 3 A B$ . They are parallel and share point $A$ , so the points are collinear.

Dividing in a ratio. If $P$ divides $A B$ in the ratio $m : n$ , then $O P = a + \frac{m}{m + n} (b - a) = \frac{n a + m b}{m + n}$ . For $A P : P B = 1 : 3$ , $O P = \frac{3 a + b}{4}$ .

Parallelograms. In $A B C D$ , $A D = B C$ , so $D = A + (C - B)$ .

Forces. The resultant is the sum of the forces. A particle is in equilibrium (stays still) when the forces add to the zero vector, so a missing force is minus the sum of the others. Find a bearing from the components: $3 i + 4 j$ is $tan^{- 1} \frac{3}{4} \approx 3 7^{\circ}$ east of north, a bearing of $03 7^{\circ}$ .

A short proof. In triangle $O A B$ with midpoints $M$ of $O A$ and $N$ of $O B$ : $M N = \frac{1}{2} b - \frac{1}{2} a = \frac{1}{2} A B$ , so $M N$ is parallel to $A B$ and half as long.

Length of a sum. $∣ p + q ∣ \leq ∣ p ∣ + ∣ q ∣$ (one side of a triangle is shorter than the other two together). They are equal only when $p$ and $q$ point the same way.

Quick summary

A vector has magnitude and direction. Write it as $A B$ , $a$ , a column, or $x i + y j$ .
$A B$ = end point minus start point.
$∣ x i + y j ∣ = x^{2} + y^{2}$ .
Add and subtract component by component. $k a$ is parallel to $a$ .
Unit vector: $a^= \frac{a}{∣ a ∣}$ .

Quiz

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Try 3 questions now

Which is a vector?
1. A a mass of 3 kg
2. B a time of 5 s
3. C a velocity of 20 m/s north
4. D a speed of 20 m/s
Show answer
C: a velocity of 20 m/s north
It has a size and a direction.
Find $∣6 i + 8 j ∣$ .
1. A 14
2. B 10
3. C 100
4. D 2
Show answer
B: 10
$36 + 64 = 10$ .
$A = (1, 2)$ and $B = (4, 6)$ . Find $A B$ .
1. A $3 i + 4 j$
2. B $5 i + 8 j$
3. C $- 3 i - 4 j$
4. D $4 i + 6 j$
Show answer
A: $3 i + 4 j$
End minus start: $(4 - 1, 6 - 2)$ .