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.
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