### Maths Magnitude

##### What is the Meaning of Magnitude?

Magnitude is a term used in various fields to describe the size, extent, or importance of something. It can be applied to physical quantities, such as force, energy, or brightness, as well as to abstract concepts, such as the severity of an earthquake or the impact of a historical event.

In mathematics, magnitude is often used to describe the size or extent of a mathematical object. For example, the magnitude of a vector is the length of the vector, and the magnitude of a matrix is the square root of the sum of the squares of its elements.

##### What is Meant by Magnitude

The magnitude of a Real number– The magnitude of a real number is usually termed the absolute value or modulus.

It is written $|x|$, and is defined by: $|x|=x,\hspace{1mm} if\hspace{1mm}x≥ 0\hspace{1mm} and \hspace{1mm}|x|=−x,\hspace{1mm} if \hspace{1mm}x<0$

Just as the magnitude of an earthquake indicates how huge the earthquake is, the magnitude of a mathematical expression tells us how significant that term is. In math, this means how far away the term is from zero or origin.

##### The Magnitude of a Vector

An object which possesses both the magnitude as well as direction is called a vector quantity. To find the magnitude of a vector, we require to calculate the length of the vector. Quantities for example velocity, force, momentum, displacement, etc. are called vector quantities.

However, speed, temperature, distance, mass, volume, etc. are known as scalar quantities. The scalar owns only the magnitude, whereas the vectors hold both magnitude and direction. Suppose, AB is a vector quantity that has a magnitude and direction both.

Then the magnitude of the vector is given by

$$\overrightarrow{|AB|}=\sqrt{(x1−x0)^2+(y1−y0)^2}$$

And the magnitude of the vector when the initial point is zero or origin:

$$\overrightarrow{|AB|}=\sqrt{x^2+y^2}$$

Example 1:

Find the magnitude of the vector $\overrightarrow{PQ}$ whose initial point, A is (2,1) and end point, B is (3,4). Using distance formula,

$$\overrightarrow{|AB|}=\sqrt{(x1−x0)^2+(y1−y0)^2}=\sqrt{(3−2)^2+(4−1)^2}=\sqrt{1^2+3^2}=\sqrt{10}$$

The magnitude of a three-dimensional vector say $xi+yj+zk$ is given by:

$$\sqrt{x^2+y^2+z^2}$$

For example to find the magnitude of $3i+2j+4k$. We would apply the same formula as

$$\sqrt{x^2+y^2+z^2}=\sqrt{3^2+2^2+4^2}=\sqrt{21}$$

Also, learn about Sequences and Series here.

##### The Magnitude of a Complex Number

Consider the complex number $a + bi$. To determine the magnitude of the complex number, calculate the modulus, which gives the distance from zero in the Argand diagram.

For the complex number $a+ib$ the magnitude is given by:

$$\sqrt{a^2+b^2}$$

For example to calculate the magnitude of the complex number $−4+3i$

$$\sqrt{(-4)^2+(3)^2}=\sqrt{25}=5$$