# 1. Vector Concepts

## Magnitude and Direction of a Vector

### On this page...

Vector Notation

Free & localized vectors

Magnitude of a Vector

Scalar Quantities

Scalar Multiplication

Opposite Vectors

Zero Vectors

Unit Vectors

A vector is a quantity that has both **magnitude** and **direction**. (Magnitude just means 'size'.)

**Examples of Vector Quantities:**

- I travel 30 km in a Northerly direction (magnitude is 30 km, direction is North - this is a displacement vector)
- The train is going 80 km/h towards Sydney (magnitude is 80 km/h, direction is 'towards Sydney' - it is a velocity vector)
- The force on the bridge is 50 N acting downwards (the magnitude is 50 Newtons and the direction is down - it is a force vector)

Other examples of vectors include:

Acceleration, momentum, angular momentum, magnetic and electric fields

Each of the examples above involves **magnitude and direction**.

**Note: **A vector is not the same as a **scalar**. Scalars have **magnitude only**.
For example, a **speed** of 35 km/h is a scalar quantity, since no direction is given. Other examples of scalar quantities are:

Volume, density, temperature, mass, speed, time, length, distance, work and energy.

Each of these quantities has magnitude only, and do not involve direction.

## Vector Notation

We will use a **bold capital letter** to name vectors. For example, a force vector could be written as **F**.

**Alternative vector notations**

- Some textbooks write vectors using an arrow above the vector name, like this:
- You will also see vectors written using matrix-like notation. For example, the vector acting from (0, 0) in the direction of the point (2, 3) can be written

`[(2),(3)]`

A vector is drawn using an **arrow**. The **length **of the arrow indicates the **magnitude** of the vector. The **direction** of the vector is represented by (not surprisingly :-) the direction of the arrow.

### Example 1 - Vectors

The displacement vector **A** has direction 'up' and a magnitude of 4 cm.

Vector **B** has the same direction as **A**, and has half the magnitude (2 cm).

Vector **C** has the same magnitude as **A **(4 units), but it has different **direction**.

Vector **D** is equivalent to vector **A**. It has the same magnitude and the same direction. It doesn't matter that **A** is in a different position to **D** - they are still considered to be **equivalent vectors** because they have the same magnitude and same direction. We can write:

A=D

**Note: **We **cannot** write **A = C** because even though **A** and **C** have the same magnitude (4 cm), they have different direction. They are not equivalent.

## Free and Localized Vectors

So far we have seen examples of "free" vectors. We draw them without any fixed position.

Another way of representing vectors is to use **directed line segments. **This means the vector is named using an **initial point** and a **terminal point**. Such a vector is called a "localized vector".

### Example 2 - Localized Vectors

A vector **OP** has initial point **O** and terminal point **P**. When using directed line segments, we still use an arrow for the drawing, with **P** at the arrow end. The length of the line OP is an indication of the magnitude of the vector.

We could have another vector **RS** as follows. It has initial point **R** and terminal point **S**.

Because the 2 vectors have the same magnitude and the same direction (they are both horizontal and pointing to the right), then we say they are equal. We would write:

OP=RS

Note that we can move vectors around in space and as long as they have the same vector magnitude and the same direction, then they are considered **equal vectors**.

## Magnitude of a Vector

We indicate the **magnitude** of a vector using **vertical lines** on either side of the vector name.

The magnitude of vector **PQ** is written |**PQ**|.

We also used vertical lines like this earlier in the Numbers chapter (where it was called 'absolute value', a similar concept to magnitude).

So for example, vector **A** above has magnitude 4 units. We would write the magnitude of vector **A** as:

| A |`= 4`

## Scalar Quantities

A **scalar quantity ** has **magnitude**, but not direction.

For example, a pen may have length "10 cm". The length 10 cm is a **scalar quantity** - it has magnitude, but no direction is involved.

## Scalar Multiplication

We can increase or decrease the magnitude of a vector by multiplying the vector by a scalar.

### Example 3 - Scalar Multiplication

In the examples we saw earlier, vector **B** (2 units) is half the size of vector **A** (which is 4 units) . We can write:

B= 0.5A

This is an example of a scalar multiple. We have multiplied the vector **A** by the scalar 0.5.

### Example 4 - Scalar Multiplication

We have 3 weights tied to a beam.

The first weight is **W _{1}** = 5 N, the second is

**W**= 2 N and the third is

_{2}**W**= 4 N.

_{3}We can represent these weights using a vector diagram (where the length of the vector represents the magnitude) as follows:

They are vectors because they all have a direction (down) and a magnitude.

Each of the following scalar multiples is true for this situation:

Since `5 = 2.5 × 2`, we can write:

W= 2.5_{1}W_{2}

Since `2 = 0.5 × 4`, we can write:

W= 0.5_{2}W_{3}

Since `4 = 0.8 × 5`, we can write:

W= 0.8_{3}W_{1}

Each of these statements is a scalar multiplication.

## Vectors in Opposite Directions

We have 2 teams playing a tug-of-war match. At the beginning of the game, they are very evenly matched and are pulling with equal force in opposite directions. We could name the vectors **OA** and **OB**.

Image source.

We can represent the tug of war using a vector diagram:

We note that the **magnitude **of each vector is the same, but they are acting in **opposite directions**. In such a case, we indicate the opposite directions by use of a **negative sign**.

So we write:

OA=−OB

## Zero Vectors

A **zero vector** has magnitude of 0. It can have any direction.

A vector may have zero magnitude at an instance in time. For example, a boat bobbing up and down in the water will have a **positive** velocity vector when moving up, and a **negative** velocity vector when moving down. At the instant when it is at the top of its motion, the magnitude is **zero**.

In the tug-of-war example above, the teams are evenly matched at a certain instant and neither side is able to move. In this case, we would have:

OA+OB=0

The 2 force vectors **OA** and **OB**, operating in opposite directions, cancel each other out.

## Unit Vectors

A **unit vector** has length **1 unit** and can take any direction.

A one-dimensional unit vector is usually written **i**.

### Example 5 - Unit Vector

In the following diagram, we see the **unit vector **(in red, labeled **i**) and two other vectors that have been obtained from **i** using scalar multiplication (2**i **and 7**i**).

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