Question

How to find the modulus of a vector?

Answer 1

Hint: In order to solve this question, we should know that any quantity which has some definite magnitude and direction is called a vector quantity and it has a specific representation and its magnitude is known as the modulus of a vector. Here, we will discuss the vector and its modulus.

Complete answer:
As we know that a vector is a quantity which has a magnitude and direction associated with it and a general vector in three dimensions is represented by $\vec A = x\hat i + y\hat j + z\hat k$ where $x,y,z$ are the components of a vector in three mutually perpendicular axes of three dimensions namely X-axes, Y-axes and Z-axes respectively.

And $\hat i,\hat j,\hat k$ are known as the three mutually perpendicular unit vectors in the direction of the X, Y and Z axes. The modulus of a vector is determined by the formula $\left| {\vec A} \right| = \sqrt {{x^2} + {y^2} + {z^2}} $. The modulus is also known as the magnitude of the vector.

For example, if we have a vector represented by $\vec A = - 3\hat i + 2\hat j + 1\hat k$ then the modulus of the given vector may be calculated using the formula
$\left| {\vec A} \right| = \sqrt {{x^2} + {y^2} + {z^2}} $ we get,
$ \left| {\vec A} \right| = \sqrt {{{( - 3)}^2} + {2^2} + {1^2}} $
$\left| {\vec A} \right| = \sqrt {14} $
which means, the magnitude of the vector will be $\sqrt {14} $ units.

Hence, to find the modulus of a general vector $\vec A = x\hat i + y\hat j + z\hat k$ use the formula $\left| {\vec A} \right| = \sqrt {{x^2} + {y^2} + {z^2}} $

Note: It should be remembered that the units of the modulus of a given vector are determined by the physical quantity which the vector represents. For example, if a vector represents speed then the units of the modulus of the vector will be of $m{s^{ - 1}}$ and scalar quantities are those which have only magnitude but no direction.