# Vectors and scalars

## Vectors and scalars

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# Vectors and scalars

A Scalar Quantity is one which has magnitude only. Examples: length, area, energy, time.

A Vector Quantity is one which has both magnitude and direction. Examples: displacement, acceleration, force.

Vectors can be represented on a diagram by an arrow, where the vector is in the same direction as the quantity it is representing.

Composition (addition) of two perpendicular vectors

• When adding two vectors, they should be arranged tail-to-tail (the arrow represents the head) and the rectangle should then be completed.
• The resultant is the line joining the two tails to the opposite corner.
• The direction is from the tails to the opposite corner.
• Mathematically, the length of the vector can be found by using Pythagoras’ Theorem.
• Mathematically, the angle can be found by using Tan q = Opp/Adj.

Experiment: To find the Resultant of Two Forces

Attach three Newton Balances to a knot in a piece of thread.

• Adjust the size and direction of the three forces until the

knot in the thread remains at rest.

• Read the forces and note the angles.
• The resultant of any two of the forces can now be shown to be

equal to the magnitude and direction of the third force.

Resolving a vector into two perpendicular Components
You have just seen that two perpendicular vectors can be added together to form a resultant.
Well let’s say we started off with the resultant. Would we be able to get back the two original vectors?

First we need to remember that for a right-angled triangle:
Sin q = Opposite/Hypothenuse, therefore Opposite = Hypothenuse x Cos q  {Opp = H Sin q}

Example
Consider a velocity vector representing a velocity of 50 ms-1, travelling at an angle of 600 to the horizontal:
The Opposite is equal to H Sin q, which in this case = 50 Cos 600 = 43 ms-1.
The Adjacent is equal to H Cos q, which in this case = 50 Sin 600 = 25 ms-1.

Now look over problem 4 and 5, page 86. Then try questions 1 – 4, page 87, followed by questions 7 and 8, page 88.

Leaving Cert Physics Syllabus: Vectors and Scalars

 Content Depth of Treatment Activities STS Vectors and Scalars Distinction between vector and scalar quantities. Vector nature of physical quantities: everyday examples. Composition of perpendicular vectors. Find resultant using newton balances or pulleys. Resolution of co-planar vectors. Appropriate calculations.

What do you get if you cross a mosquito with a rock climber?
You can't cross a vector with a scalar!
Boom Boom!

Exam questions

• [2003]

Give the difference between vector quantities and scalar quantities and give one example of each.

• [2006 OL]

Force is a vector quantity. Explain what this means.

• [2003]

A cyclist travels from A to B along the arc of a circle of radius 25 m as shown.

• Calculate the distance travelled by the cyclist.
• Calculate the displacement undergone by the cyclist.

• [2004]

Two forces are applied to a body, as shown. What is the magnitude of the resultant force acting on the body?

• [2003]

Describe an experiment to find the resultant of two vectors.

Exam solutions

• A vector has both magnitude and direction whereas a scalar has magnitude only.
• A vector is a quantity which has magnitude and direction.

•
• The displacement is equivalent to one quarter of the circumference of a circle = 2πr/4 = 25π/2

= 12.5π = 39.3 m.

• Using Pythagoras: x2 = 252 + 252 Þ x = 35.3 m. Direction is NW
• R2 = F12 + F22  Þ  R2 = 52 +122

R = 13 N
The marking scheme didn’t look for direction, but it should have, particularly since this is a vectors question and force is a vector. q = tan-1 (5/12).

•
• Attach three Newton Balances to a knot in a piece of thread.
• Adjust the size and direction of the three forces until the
• knot in the thread remains at rest.
• Read the forces and note the angles.
• The resultant of any two of the forces can now be shown to be

equal to the magnitude and direction of the third force.

Source : http://www.thephysicsteacher.ie/LC%20Physics/Student%20Notes/8.%20Vectors%20and%20Scalars.doc

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