Scalars and Vectors A scalar is a physical quantity that has only magnitude but no direction. Examples: speed, distance, time, mass, volume. A vector is a physical quantity that has both magnitude and direction. Examples: velocity, displacement, acceleration, force, weight.
Addition of scalars Two scalars can be added using simple mathematical addition e.g. 2 h + 3 h = 5 h
Addition of vectors Vectors cannot be added using simple mathematics because they have a direction also. To add two vectors we need to use the head- to-tail rule or the parallelogram rule. We represent a vector by an arrow.
head
tail
The arrow-head shows the direction while the length of the line gives the magnitude of the vector.
Head-to-Tail Rule Given two vectors 5 N and 10 N at an angle of 30o. Note that they are joined tail to tail. Two vectors can only be added if they are joined by head-to-tail. Now follow the steps to add them up. 5N 30o 10 N
Step 1: Join the two vectors by head to tail by drawing a line parallel to the 5 N force in front of the 10 N force as shown. 5N 30o
150o
5N
10 N The tail of the 5 N force is joined to the head of the 10 N force. This is the HEAD-TO-TAIL method. The angle between the two vectors now will be 180 o – 30o = 150o (the two angles are always supplementary).
Step 2: Draw a scale diagram. Choose a suitable scale. Here I will take 1 cm : 1 N So 5 N = 5 cm and 10 N = 10 cm.
150o
5 cm
10 cm
Step 3: Draw the resultant as shown and measure it in cm. Then convert it into Newtons ( N) by multiplying it with the scale i.e. by 1.
Resultant in cm
10 cm
5 cm
Step 4: to find the direction of the resultant measure the angle between the tail of the resultant and the nearest vector. Resultant
Îą 10 N
5N
Direction of resultant: Îą with 10 N force.
Parallelogram rule Step 1: Draw a parallelogram as shown.
5N
30o
resultant
10 N
Step 2: Draw a scale diagram and measure the diagonal of the parallelogram. This will give the resultant of the two forces. Step 3: the direction of the resultant is the angle between the tail of the resultant and the nearest vector. You can take either the angle with the 10 N force or the 5 N force.
By Shafaq Hafeez shafaq@physics.com.pk