Video Lesson

Lesson objective

Dear learners,

At the end of the lesson you will be able to

• Describe the difference between vector and scalar quantities.
• List down the common vector quantities in our everyday life.
• Discuss geometric representation of vectors.
• Give the definitions of the different types of vectors.

Brainstorming Question

Why do we need to know the direction of some quantities in order to have a complete sense of them? If a girl told you that the velocity of a delivery truck was 50 km/h, has she fully expressed the motion of the truck?

Key Terms and Concepts

LESSON Presentation

In mathematics and physics, we have physical quantities which can be categorized in two ways, namely

• Scalar Quantity
• Vector Quantity

1.Scalar Quantity Definition

• The physical quantities which have only magnitude are known as scalar quantities.
• It is fully described by a magnitude or a numerical value.
• Scalar quantity does not have directions.
• In other terms, a scalar is a measure of quantity.

For example,

if I say that the height of a tower is 15 meters, then the height of the tower is a scalar quantity as it needs only the magnitude of height to define itself. Let’s take another example, suppose the time taken to complete a piece of work is 3 hours, then in this case also to describe time just need the magnitude i.e. 3 hours.

Examples of scalar quantities are

Mass, speed, distance, time, energy, density, volume, temperature, distance, work and so on.

2.Vector Quantity Definition

• The physical quantities for which both magnitude and direction are defined distinctly are known as vector quantities.

For example,

A boy is riding a bike with a velocity of 30 km/hr in a north-east direction. Then, as we see for defining the velocity, we need two things, i.e. the magnitude of the velocity and its direction. Therefore, it represents a vector quantity

Examples of Vector quantities are

Displacement, acceleration, force, momentum, weight, the velocity of light, a gravitational field, current, and so on.

Representation of vectors

• The representation of vector is done by using the arrow.
• We know that an arrow contains a head and a tail.
• The head of the arrow denotes the direction of the vector..
• The representation of vector is done by a directed line segment.
• It is an arrow that has a head and a tail. here,
• The starting point of the vector is called its tail (or) the initial point of the vector.
• The ending point of the vector is called its head (or) the terminal point of the vector.
• The head of the vector shows its direction.
• The direction of the vector is the angle made by it with a reference line.

Types of vectors

• Parallel vectors: Vectors that are in the same direction are said to be Parallel, Figure 2.4a.
• Antiparallel vectors: When vectors have opposite directions, whether their magnitudes are the same or not, they are Antiparallel, Figure 2.4b.
• Equal vectors: If two vectors representing the same quantity have the same magnitude and the same direction, they are Equal vectors, Figure 2.4c, no matter where they are located in space.
• Collinear vectors: These are vectors that lie along the same line or parallel lines. Two vectors are collinear if they are parallel to the same line irrespective of their magnitudes and direction.
• Co-planar vectors: Three or more vectors lying in the same plane or parallel to the same plane are known as co-planar vectors.
• Zero vector: A zero vector is a vector when the magnitude of the vector is zero and the starting point of the vector coincides with the terminal point.
• Orthogonal vectors : Vectors that are perpendicular to one another.
• A unit vector: A vector which has a magnitude of unit length is called a unit vector.
• Negative of a vector as a vector having the same magnitude as the original vector but the opposite direction.
• The negative of vector A is denoted by
• A = –B (“Vector B is negative of vector A”)

Examples

1. Given the vector P = (2, 4), determine the negative of P.

Solution

By definition, the negative of a vector has the same magnitude as the reference vector’s opposite direction. In this case, the reference vector is P, and its direction is 2 points to the right along the x-axis and 4 points upward along the y-axis. Thus, to find P’s negative vector, we keep the same magnitude and multiply the reference vector P by -1. This gives us:

–P = (-2,-4)

2. Determine the value of n for which the two vectors A = (-5, -1, 3n) and B = (-5, -1, -9) are the negatives of each other.

Solution

We know that two vectors are equal if their magnitudes are the same and their directions are opposite to each other. We use this to determine the value of the unknown n as follows:

A = – B  => (-5, -1, -3n) =  – (-5, -1, -9)

By setting the corresponding components equal to each other, we get:

-5 = 5, -1 = 1, and  3n = 9

-3n =  -9

By simplifying the above equation, we get:

n = 3

Thus, when n = 3, the two vectors A and B are the negatives of each other.