Magnitude of a Vector Formula

The magnitude of a vector formula helps to know the length of a vector, a quantity that has a definite direction along with magnitude.

There are two types of quantities – Scalar and Vector. The scalar quantities only have a magnitude while vector quantities refer to those which have direction along with magnitude.

The magnitude of a vector, also known as modulus of vector, is the measurement of length between the initial point A and the terminal point B of a vector AB and is expressed as |AB^→|

Hence, it can be said that the vectors have the same magnitude although their direction is different (opposite to each other).

If there is a vector AB where coordinates of A and B are (x₁, y₁, z₁) and (x₂, y₂, z₂) respectively, then its magnitude denoted by AB is given by the formula AB = √(x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²

In particular, for A (0, 0, 0) and B (x, y, z), it will be

r^→ = |AB^→| = xi^ + yj^ + zk^

And |AB^→| =  √x² +y² + z², where r is the position vector of the point B and xi^, yj^, zk^ are called vector components of r.

Solved Examples  

Question 1. Find the magnitude of the vector MN joining the points M (1, 2, 3) and N (0, -1, 5).

Solution:  Here, x₁ = 1, y₁ = 2, z₁ = 3, x₂ = 0, y₂ = -1, z₂ = 5

Therefore, magnitude =|MN| = √(x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²

Or magnitude = √(0 – 1)² + (-1 – 2)² + (5 – 3)²

Or magnitude = √( – 1)² + (-3)² + (2)²

Or magnitude = √(1 + 9 + 4)

Or magnitude = √14

Question 2. What will be the value of x for which x(i^+j^+k^) is a unit vector?

Solution: Since it is given to be a unit vector, its magnitude is 1, therefore, | x(i^+j^+k^) | = 1

Applying the magnitude of vector formula,

√x² + x² + x² = 1

Or √3 x² = 1

Or √3x = 1

Or x = 1/√3 or -1/√3