Vectors and Scalars

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Contents 1 Scalar 2 Vector 3 Addition 4 Resolving a vector into two perpendicular Components 5 Experiment

Scalar

• Definition: A Scalar Quantity is one which has magnitude only
• Examples:

1. Length
2. Area
3. Energy
4. Time

Vector

• Definition: A Vector Quantity is one which has both magnitude and direction
• Examples:

1. Displacement
2. Velocity
3. Acceleration
4. Force

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

Addition

Composition (addition) of two perpendicular vectors:

• When adding two vectors, they should be arranged tail-to-tail (where 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 (as in diagram below)
• 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 = Opp/Adj.

Resolving a vector into two perpendicular Components

Two perpendicular vectors can be added together to form a resultant (as proven in the experiment below). Well let’s say we started off with the resultant. Would we be able to get back the two original vectors?

The answer is yes, and to do so we make use of the following:

If the vector (of magnitude H) makes an angle with the horizontal, then the Opposite is equal to H Sin , and the adjacent vector is equal to H Cos .

• Proof
  

Cos = Adjacent/Hypothenuse,
therefore Adjacent = Hypothenuse x Cos
{Adj = H Cos }

Sin = Opposite/Hypothenuse,
therefore Opposite = Hypothenuse x Cos
{Opp = H Sin }

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

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