Vectors And Scalars

In this lesson I will describe vectors and scalar quantities , their differences and different types of vectors including unit vectors.

Some physical quantities can be completely described with a single number such as time, weight, length etc. These quantities are called scalar quantities. But in some cases only a single number is not enough. Suppose, your home is 5 km north from your school. Someone asked you about your home’s location, and you said that your house is 5 km away from your school. If he wants to find your home, there are infinitely many options for him. He may go 5 km east or 5 km west or in any other direction. The only way he may find your home is if he goes 5 km north.

So, in this case “5 km” is not enough to describe the distance between your house and your school. You need to specify the direction too. These types of quantities are called vector quantities. In this case, the value combined with the direction is called displacement. Another example of vector quantity is force. A force has both a magnitude and a direction.

A vector is represented by an arrow. The length of the arrow represents the magnitude or the value of the vector and the direction of the arrow is the direction of the vector.

A vector can be represented in three ways. First, with two letters and an arrow over them, where the first letter is the starting point and the second letter is the ending point. Second, with a single letter with an arrow over it. Third, with a bold letter. The magnitude or the length of the vector is represented by an absolute symbol with the vector.

Vector Equality

Two vectors can be equal if both of them have the same length and the same direction.

In the previous image, both A and B have the same length and the same direction. Hence, A and B vectors are equal. A and D also have the same length, but their directions are opposite. So, A and D are not equal vectors. and C have the same directions but their lengths are different. So, these two vectors are not equal.

Vector Multiplication With Scalar Quantity

If we multiply a vector with a scalar quantity, the result will be another vector. If we multiply the vector with a positive number, then the direction will be unchanged and the length will be multiplied with the positive number.

A has a length of 3 . Multiplying it with 2 multiplies its length 2 times. But the direction remains unchanged.

Multiplying a vector with a negative number changes its direction and multiplies the length.

Unit Vector

A unit vector is a vector, whose length is 1 unit. Let, B is a vector with a length of 3. “b” is another vector with a length of 1. Both B and b have the same direction. We call b , an unit vector in the direction of B.

If we multiply b with 3, then its length will be the same as B. As both b and B have the same direction, the two will then be equal.

Thus, dividing a vector with its length gives a unit vector in the same direction. An unit vector is often represented by a hat ‘ ^ ‘over a letter.