Scalar and Vector Quantities – SS2 Physics Lesson Note (First Term)

A scalar quantity has magnitude only, such as distance, speed, mass and energy. A vector quantity has both magnitude and direction, such as force and weight. A vector is drawn as a line whose length shows its size and whose arrowhead shows its direction. Vectors acting together can be combined into a single vector called the resultant.

Behavioural objectives

By the end of the lesson, students should be able to:

• distinguish between scalar and vector quantities and give examples of each;
• represent a vector graphically;
• find the resultant of two vectors acting in the same direction, in opposite directions and at 90° to each other;
• state the parallelogram law of vector addition;
• resolve a vector into two components;
• find the resultant of more than two vectors.

What are scalar and vector quantities?

Physical quantities are divided into scalar and vector quantities.

A scalar has only magnitude (size), e.g. distance, speed, temperature, volume, work, energy, power, mass etc.

A vector quantity has both magnitude and direction, e.g. force, weight, magnetic flux, electric fields, gravitational fields etc.

How is a vector represented?

A vector quantity can be represented graphically by a line drawn so that the length of the line shows the magnitude of the quantity. The direction of the vector is shown by the arrowhead.

How are vectors added and subtracted?

Two or more vectors acting on a body in a specified direction can be combined to produce a single vector having the same effect. The single vector is called the resultant.

Vectors in the same direction

Two forces Y and X with magnitudes of 3 N and 4 N respectively, acting along the same direction, will produce a resultant of 7 N (the algebraic sum of the two vectors).

Vectors in opposite directions

If Y and X act in opposite directions, the resultant will be 1 N.

Vectors inclined at 90° to each other

If the two vectors are inclined at 90° to each other, Pythagoras’ theorem is used.

R² = X² + Y²
R² = 4² + 3²
R² = 16 + 9
R² = 25
R = √25
R = 5 N

The direction of the resultant is found as follows:

tan θ = Y/X
θ = tan^-1(Y/X)
θ = tan^-1(3/4)
θ = tan^-1(0.75)
θ = 36.9°

Vectors inclined at other angles

If the two vectors are inclined at an angle less than 90° or more than 90°, the resultant is obtained by using the parallelogram law of vector addition.

The parallelogram law of vector addition states that if two vectors are represented in magnitude and direction by adjacent sides of a parallelogram, the resultant is represented in magnitude and direction by the diagonal of the parallelogram drawn from the common point.

What is resolution of vectors?

A single vector can be resolved into two vectors called components. A vector F represented as the diagonal of a parallelogram can be resolved into its components, which are then taken as the adjacent sides of the parallelogram.

How do we find the resultant of more than two vectors?

To find the resultant of more than two vectors, resolve each vector in two perpendicular directions. Then add all the horizontal components, X, and all the vertical components, Y.

For example, consider four forces acting on a body as shown below.

Add all the resolved horizontal components (Figure 1):

X = F₁ cos θ₁ + (-F₂ cos θ₂) + (-F₃ cos θ₃) + F₄ cos θ₄

Y = F₁ sin θ₁ + F₂ sin θ₂ + (-F₃ sin θ₃) + (-F₄ sin θ₄)

Then find the resultant (Figure 2):

R² = X² + Y²
R = √(X² + Y²)

The direction α is given by tan α = Y/X.

Evaluation

1. What is the difference between a scalar quantity and a vector quantity?
2. Two forces of 3 N and 4 N act in opposite directions. What is the resultant?
3. State the parallelogram law of vector addition.
4. Describe how to find the resultant of more than two vectors.

Assignment

1. List three scalar quantities and three vector quantities from this lesson.
2. Two forces of 6 N and 8 N act at 90° to each other. Calculate the resultant.