A** vector** is a quantity
with both magnitude and direction.

A **scalar **is a quantity
with magnitude only.
**Examples of vectors**
in the physical world are: velocity, acceleration, force, weight.
**Examples of scalars**
in the physical world are: speed, time, and mass.

A vector may be represented
diagrammatically by a directed line segment.

The length of the line
segment represents the magnitude (norm) of the vector.

The direction of the vector
is represented by the direction of the line segment.

**Examples**

A velocity of 20 m/s due
north may be represented as follows: 20 m/s

A force of 30 Newton’s
inclined at an angle of 30° to the horizontal may be represented as
follows:

*Addition of vectors*

Two vectors may be added
using the** Triangular Law of addition.**

Let’s say we wish to find
the* ***resultant **of two forces, one a force of 20N due north,
the other a force of 30 N inclined at an angle of 30° to the horizontal.

**R** above represents
the single force, which would have the same effect as the two original
forces if they were applied to a particle.

We may find the magnitude
of R, i.e. ½ R½ using the **Cosine Rule **(Tables page
9)

½ R½^{2}
= 20^{2} + 30^{2} - 2.20.30.cos120° = 400 + 900 - 1200(-½)
= 1900

½ R½ = =10N

The dir. of R** **may
be found using the** Sine Rule
**(Tables page 9)

Þ

36° 35/

In **Relative Velocity**
questions we are required to subtract vectors.

Subtraction is the addition
of the negative of a vector (i.e. the vector with its dir. reversed)

Suppose we have the following
vectors:

is a velocity of 20 km.hr due east.

is a velocity of 15 km/hr due north.

We wish to find

½½^{2}
= 20^{2} + 15^{2} = 625

½½ = 25 km/hr

tan a = ® a = 36° 52/

**Resolving a vector into
two perpendicular components**

In many topics in Applied
Mathematics it is necessary to split up (resolve) a vector into two components.
For example in studying projectile motion we need to resolve the initial
velocity and the acceleration due to gravity into components.

For a projectile on the horizontal plane these two components are in the dir. of the X-axis and the dir. of the Y-axis. With a projectile on the inclined plane the directions will be parallel and perpendicular to the plane.

We let
= a unit vector in the dir. of the X-axis

= a unit vector in the dir. of the Y-axis

i.e. ½½
= 1 and the dir. of is
due east

and ½½
= 1 and the dir. of is due north.

Vector
Resolved

**The magnitude and dir.
of a vector written in terms ofand **

If

then

and the angle at which
is inclined to the horizontal is given by

**The Scalar (dot) product
of two vectors**

If we have two vectors
then

where
is the angle between

It is important to note that if are perpendicular then = 90° and cos 90° = 0