A note on Vectors

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:

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 R½2 = 202 + 302 - 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

½½2 = 202 + 152 = 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

Hence