Scalars and Vectors

Scalars and Vectors

Up until this unit, there are certain concepts in Physics that are used either interchangeably or downright incorrectly in "everyday" language. 

Scalars and Vectors is a new concept that will be used continuously throughout Physics all the way up to Advanced Higher and beyond. The difference between a Scalar and Vector is as follows :-

1. Scalar - Has a magnitude only (eg 45 m, 45 ms-1, 45 kg)

2. Vector - Has a magnitude and direction (eg 45 m North, 45ms-1 East) 

Speed and Velocity

Up to now these two words have been seen as interchangeable. This is no longer the case. The following must now be used when describing motion :-

1. Speed - Scalar quantity (magnitude only)

2. Velocity - Vector quantity (magnitude and direction) 

Distance and Displacement

Up to now, only distance has been discussed. Now, its Vector equivalent will also be used :-

Distance - Scalar quantity - How far the object actually moved 

Displacement - Vector quantity - How far is the end point from the start

When to use which?

During calculations, you must remain consistent with either use of Scalar or Vectors. For example, using Distance will give an object's Speed, whereas using Displacement will give an object's Velocity.

Vectors and Direction 

As stated above, in order to give a full description to a Vector, a direction must be given. There are several methods for giving a direction that can be used in Physics, with the two main methods described below.

1. Method one - Angle from a reference point

2. Method two - Three figure Bearing

Example 1 -

What is the displacement of a runner who travels the route A-B-C ? 

Using Pythagoras to find the magnitude :-

displacement  =  square root of (42 + 32

                          =  5 km 

Using Trigonometry to find the direction :-

Tan-1 ( θ )  = (4 / 3)

                               θ   = 53°

Note - This value of 53° is not enough information to give a direction clearly. You must give a position and direction of this angle to give the information clearly. 

Method 1 - Displacement 5 km at an angle of 53° East of North

The above method is useful as it requires no further work to give an answer. This method, however, risks getting confused with direction (eg is it East of North or North of East ?). 

In order to avoid this confusion, three figure bearings can be used. A three figure bearing is the angle clockwise from North. This prevents the problems of direction in any answers. The diagram below shows the compass points given as three figure bearings :-

Method 2 - Displacement 5 km at a Bearing of 053 

Note - Either method is acceptable, unless the question specifically asks for a particular method.


Direction Conventions for Calculations

When working with Vectors, direction is important. In order to perform calculations based upon Vectors a decision has to be made as to which direction is a positive direction and which is a negative direction. 

By convention, In the Higher Physics course, we normally use the following :-

Positive direction - Upwards or to the Right

Negative direction - Downwards or to the Left

Note - As the acceleration due to Gravity acts downwards, in calculations the value -9.8 ms-2 would normally be used. This will be used extensively in the Equations of Motion section of the course.