Scalars and Vectors


Scalar and Vector Quantities

Scalar values describe the magnitude or size of a variable only. Values which are commonly only described as scalar quantities include:

  • distance
  • speed
  • time

Vector values describe the magnitude or size and the direction of a variable. Values which are commonly only described as vectors include:

  • position
  • displacement
  • velocity
  • acceleration
  • force

Describing Direction

When dealing with vector problems in one dimension there are a number of different conventions which can be used to describe the direction of a vector. The convention used in an answer should usually be in the same format as expressed in the problem. Some common conventions include:

  • forwards, backwards
  • up, down
  • left, right
  • east, west
  • north, south

Other methods are used for describing vectors in two dimension and these will be discussed in a later section.


Sign Conventions

With one dimensional problems the descriptive directions can be converted to positive (one direction) and negative (the opposite direction) to assist with mathematical problems.

For example, if two cars passed each other on a highway, with car A travelling east at 10 m/s and car B travelling west at 15 m/s. We could denote east as being positive and say that: car A is travelling at 10 m/s and car B travelling at -12 m/s.


Relative Motion

When two cars, car A and car B, travelling in opposite directions pass each other on a road, the motion of each vehicle may be described differently depending how each vehicle is observed. For example, the motion of car A could be described from the perspective of the driver in car B, or by a stationary observer on the side of the road. These observed differences are known as relative motion. Relative motion is observed when an event is observed from different frames of reference.

A classic analogy is one where a sailor drops a rock from the mast of a boat travelling along the shore. An observer on the shore and the sailor will observe different motion, or a different trajectory for the rock. This is because the sailor and the observer on the shore are observing the rock fall from different frames of reference. The sailor will observe the rock fall straight down and the observer on the shore will observe the rock follow a parabolic trajectory as shown below.

It is important to note that both observations are correct as no frame of reference is more correct than any other frame of reference.


Relative Velocity

Imagine you are stationary on the side of the road and a car drives pass you at 10 m/s in a northerly direction. You would describe the cars velocity as 10 m/s north. To the driver of the car, it would appear as though you were moving in the opposite direction. The driver would describe your motion as 10 m/s south. This is a description of relative velocity.

  • Relative velocity can be calculated as:

velocity of object A relative to object B = velocity of object A – velocity of object B

  • The notation for relative velocity:

{ v }_{ AB }\:=\:{ v }_{ A }-{ v }_{ B }

  • It is important to note that we can not subtract vectors as outlined in the equation. To overcome this we can add the negative vector: (a negative vector has the same magnitude but opposite direction)

{ v }_{ AB }\: =\: { v }_{ A }+(-{ v }_{ B })


Example 1:

Describe the vector illustrated below using:

a) the convention shown

b) the sign convention shown

Answers:

a) 25m east

b) + 25m


Example 2:

A red car (R) is travelling north at 17 m/s and a blue car (B) is travelling south at 9 m/s. Calculate the speed of the blue car relative to the red car.

Answer:

{ v }_{ RB }\:=\:{ v }_{ R }-{ v }_{ B }

{ v }_{ RB }\: =\: { v }_{ R }+(-{ v }_{ B })

*denoting north as positive

{ v }_{ RB }\: =\: 17\:+\:9

{ v }_{ RB }\: =\:26 m/s (26 m/s north)

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