# Scalar and Vector Quantities

Aim: Resolve, add and subtract vectors in one plane

A scalar is any quantity with a magnitude, but no direction.

A vector is any quantity with both magnitude and direction.
Vectors can be represented as follows to distinguish between the two:

Vectors are represented by a line with an arrow:

Length= magnitude; angle = direction

### Direction:

Expressed in full circle bearing (true bearing), a clockwise angle from north

OR

Expressed as a quadrant bearing, the angle between two cardinal directions and the line

### Vector Components

It is possible to represent a vector by splitting it into two perpendicular components. (Usually vertical and horizontal).

### Vector Scaling

Multiplying a vector multiples the magnitude but does not change the direction.

If multiplied by a negative, the direction is reversed.

Graphical:

1. Draw a reference frame
2. Draw the first vector
3. Draw the second vector, so that the start of the second vector is the head of the first
4. Repeat for all n vectors
5. Draw the resultant vector from the tail of the first vector to the head of the last one.

Note that this should be to scale.

Algebraic:

1. Calculate the perpendicular components of each vector
2. Add similar components of the different vectors (add horizontal to horizontal etc.)
3. Using the new values, draw a vector diagram
4. Use Pythagoras and trigonometry to determine the resultant vector

### Vector Subtraction

Subtracting by a vector is the same as adding its negative.

That is

vector subtraction = fa – fb

= fa + (-fb)

1. Flip the vector you are subtracting by. Magnitude remains the same, but direction is reversed
2. The rest is the same as vector addition