Physics and Physical Measurement

Topic 1.3 Scalars and Vectors

Scalars Quantities

Scalars can be completely described by magnitude (size)

Scalars can be added algebraically They are expressed as positive or negative

numbers and a unit examples include :- mass, electric charge,

distance, speed, energy

Vector Quantities

Vectors need both a magnitude and a direction to describe them (also a point of application)

When expressing vectors as a symbol, you need to adopt a recognized notation

e.g. They need to be added, subtracted and multiplied in

a special way Examples :- velocity, weight, acceleration,

displacement, momentum, force

Addition and Subtraction

The Resultant (Net) is the result vector that comes from adding or subtracting a number of vectors

If vectors have the same or opposite directions the addition can be done simply

same direction : add opposite direction : subtract

Co-planar vectors

The addition of co-planar vectors that do not have the same or opposite direction can be solved by using scale drawings to get an accurate resultant

Or if an estimation is required, they can be drawn roughly

or by Pythagoras’ theorem and trigonometry Vectors can be represented by a straight line

segment with an arrow at the end

Triangle of Vectors

Two vectors are added by drawing to scale and with the correct direction the two vectors with the tail of one at the tip of the other.

The resultant vector is the third side of the triangle and the arrow head points in the direction from the ‘free’ tail to the ‘free’ tip

Example

a b+ =

R = a + b

Parallelogram of Vectors

Place the two vectors tail to tail, to scale and with the correct directions

Then complete the parallelogram The diagonal starting where the two tails

meet and finishing where the two arrows meet becomes the resultant vector

Example

a b+ =R = a + b

More than 2

If there are more than 2 co-planar vectors to be added, place them all head to tail to form polygon when the resultant is drawn from the ‘free’ tail to the ‘free’ tip.

Notice that the order doesn’t matter!

Subtraction of Vectors

To subtract a vector, you reverse the direction of that vector to get the negative of it

Then you simply add that vector

Example

a b- =

R = a + (- b)-b

Multiplying Scalars

Scalars are multiplied and divided in the normal algebraic manner

Do not forget units! 5m / 2s = 2.5 ms-1

2kW x 3h = 6 kWh (kilowatt-hours)

Multiplying Vectors

A vector multiplied by a scalar gives a vector with the same direction as the vector and magnitude equal to the product of the scalar and a vector magnitude

A vector divided by a scalar gives a vector with same direction as the vector and magnitude equal to the vector magnitude divided by the scalar

You don’t need to be able to multiply a vector by another vector

Resolving Vectors

The process of finding the Components of vectors is called Resolving vectors

Just as 2 vectors can be added to give a resultant, a single vector can be split into 2 components or parts

The Rule

A vector can be split into two perpendicular components

These could be the vertical and horizontal components

Vertical component

Horizontal component

Or parallel to and perpendicular to an inclined plane

These vertical and horizontal components could be the vertical and horizontal components of velocity for projectile motion

Or the forces perpendicular to and along an inclined plane

Doing the Trigonometry

Sin = opp/hyp = y/V

Cos = adj/hyp = x/V

V

y

x

Therefore y = Vsin In this case this is the vertical component

Therefore x = Vcos In this case this is the horizontal component

V cos

V sin

Quick Way

If you resolve through the angle it is cos

If you resolve ‘not’ through the angle it is sin

Adding 2 or More Vectors by Components First resolve into components (making sure

that all are in the same 2 directions) Then add the components in each of the 2

directions Recombine them into a resultant vector This will involve using Pythagoras´ theorem

Question

Three strings are attached to a small metal ring. 2 of the strings make an angle of 70o and each is pulled with a force of 7N.

What force must be applied to the 3rd string to keep the ring stationary?

Answer

Draw a diagram

7N 7N

F

70o

7 sin 35o7 sin 35o

7 cos 35o + 7 cos 35o

Horizontally 7 sin 35o - 7 sin 35o = 0

Vertically 7 cos 35o + 7 cos 35o = F F = 11.5N And at what angle? 145o to one of the strings.