Topic 1 Physical Measurement

1.3 Vectors and Scalars

Vectors and Scalars

In Physics some quantities have values that depend on direction and some do not.

A quantity that has a direction associated with it is called a vector.Example velocity.

A quantity that has no direction is called a scalar.Example mass

For a vector quantity a + or - sign shows direction

For a scalar quantity a - sign is (usually) meaningless

Vectors and Scalars

Sort the list of quantities below into a table of vectors and scalars.MassDisplacement

TimeEnergy

VelocitySpeed

DistanceAcceleration

Amount of SubstanceForce

Representing Vectors

A vector is generally represented in equations by using an underline, or by an over-arrow.Some textbooks use bold fonts to show vectors.Example:

Representing Vectors

Vectors are represented in diagrams by an arrow (often drawn to a scale) pointing in the correct direction but attached by its tail to the point at which it acts.Example

W=45N

F=20N

Combining Vectors

Often in Physics we need to find what the total effect of a group of vectors is.

The one vector that can be used to cause the same effect as the group is called the resultant vector.

Often (but not always) the resultant is drawn as a double headed arrow.

Adding Vectors

In order to add vectors graphically we need to draw them to scale following some basic rules.Decide on a scale and stick to it!

Draw one vector and label it if necessary.

Draw the next vector starting at the head of the first one.

Carry on doing this until all vectors are drawn.

Join the starting point to the end point and mark this as the resultant.

Measure the resultants length and convert this using your scale into the correct units.

Measure the direction of the resultant from a sensible datum.e.g. above horizontal, right of vertical, below the direction of vector 1.

Adding Vectors

Example: What is the resultant of a 7N horizontal force to the right and a 12N force vertically down?

Scale 1cm=1N

13.9cm = 13.9N

60o

Answer: 13.9N @ 60o below Horizontal

Subtracting Vectors

Subtracting vectors follows exactly the same process as addition.Example: two vectors a and b are defined as
a = 5ms-1 horizontal to the right
b = 10ms-1 to the right.

What is r = a b ?

Subtracting Vectors

a b = a + (-b)-b is the same magnitude (size) as b but acts in the opposite direction to b.

a

b

-b

a

-b

r

r = a + b

r = a - b

-b

r

Combining Vectors Mathematically

Consider any vector a.It can be resolved (broken down) into 2 orthogonal components ax and ay.

This is done using the sine and cosine identies

a

ax

ay

Combining Vectors Mathematically

The same process can also be used to resolve vector b into its components.

To find the resultant r is then simply a matter of summing the x components and the y components.rx = ax + bx + ry = ay + by + ...

These resultant components can then be combined using Pythagoras and tangent to find the magnitude and direction.

Combining Vectors Mathematically

When working with large numbers of vectors it is often useful to represent each vector as a column matrix.

This then does not require x and y subscripts and can be easier to follow.This is almost essential if working with x, y and z!

Example

John Kennedy