What is the magnitude?

The magnitude is a term associated with the vector. The magnitude of a vector is the total length of a vector. The magnitude of a vector is also known as the absolute value. If is any vector, then the magnitude of it is represented as $\left|x\right|$. For a vector having coordinates in X, Y, and Z directions, the individual measures in each direction can be represented using the magnitude of the vector. The magnitude of a vector represents the distance between two points in space.

When the magnitudes are compared using the logarithmic scale, the magnitudes are known as logarithmic magnitudes. The magnitudes of the loudness of sound, the brightness of a star, and the measurement of the intensity of an earthquake are all measured as logarithmic magnitudes.

Vectors

A vector is an element with both direction and magnitude. A vector is represented using an arrow, where the arrow represents the direction of the vector and its length is represented using the magnitude. The set of vectors in space is known as vector space. The vectors can be added, subtracted, multiplied, and divided in a vector space. The image below shows a vector AB.

Any vector $x$ in a three-dimensional space can is represented as, $\overrightarrow{x } = a_x\overrightarrow{i} + a_y\overrightarrow{j} + a_z\overrightarrow{k}$.

Where, $\overrightarrow{i}, \overrightarrow{j}, and \overrightarrow{k}$ are the unit vectors that represent the direction of a vector and $a_x, a_y, and a_z$ are the components of the vector in $X, Y, and Z$ directions, respectively. 

Types of vectors

• A zero vector has a magnitude equal to zero and no direction.
• A unit vector has a magnitude equal to 1. It is used to represent the direction of a vector.
• A position vector is used to represent the direction and position of a vector in a three-dimensional space. It is also known as a location vector.
• A negative vector is a term used for multiple vectors. A vector is known as a negative of another vector when it has the magnitude same as that of another vector, but opposite direction.
• An equal vector is also a term used for multiple vectors. The vectors are said to be equal when they have the same direction as well as the same magnitude.
• A parallel vector is also a term associated with multiple vectors. The vectors are said to be parallel when the vectors have the same direction. The magnitude of the vectors may or may not be the same.
• The co-initial vectors is the term used when multiple vectors have the same initial point.
• The vectors are said to be orthogonal vectors when two or more vectors have an angle of 90 degrees between them.

The vectors are distinguished based on their magnitude, direction, and relationships with other vectors. The various types of vectors are - zero vectors, unit vectors, position vectors, negative vectors, equal vectors, parallel vectors, co-initial vectors, and orthogonal vectors.

Finding the magnitude of a vector

The magnitude of a vector is generally calculated for the vector in both two-dimensional and three-dimensional planes. Following are the steps involved in calculating the magnitude of a vector-:

• Identification of the components is done in all the dimensions.
• The squares of the magnitudes of the components in all the dimensions are calculated.
• The terms thus obtained are then added.
• The square root of the obtained sum is taken.

The magnitude of real numbers and complex numbers

Real numbers are numbers that contain both rational and irrational numbers. 

If a is any real number, then the magnitude of a will be given as $\left|a\right|$.

$\left|a\right| = a$ if, $a\ge 0$ and $\left|a\right| = -a$ if $a<0$.

Imaginary numbers are the numbers that are formed using real numbers. When the real numbers are multiplied by the imaginary number i, imaginary numbers are formed. The value of i = $\sqrt{-1}$

If b is any complex number having the value $b=x+iy$, then, the magnitude of b can be given as,

$\left|b\right| = \sqrt{x^2 + y^2}$

The magnitude of a vector in a two-dimensional plane

Consider a vector $\overrightarrow{A}$ having the endpoints $(x_1,y_1)$ and $(x_2,y_2)$. The magnitude of the vector $\overrightarrow{A}$ is calculated using the following formula:

$\left|\overrightarrow{A}\right| = \sqrt{{(x_2 - x_1)}^2 + {(y_2 - y_1)}^2}$

Consider a vector $\overrightarrow{B}$ having the endpoints $(0,0)$ and $(x,y)$. The magnitude of the vector $\overrightarrow{B}$ is calculated using the following formula:

$\left|\overrightarrow{B}\right| = \sqrt{{(x-0)}^2 + {(y-0)}^2}$

$\left|\overrightarrow{B}\right| = \sqrt{x^2 + y^2}$

The direction of a vector

The direction of a vector is the measurement of the angle that the vector makes with the horizontal line. Consider a vector $\left|\overrightarrow{C}\right|$ having the endpoints $(x_1,y_1)$ and $(x_2,y_2)$.

The direction of the above vector is calculated as below:

$\theta = {\tan }^{-1}\left(\frac{y_2 - y_1}{x_2 - x_1}\right)$

where, $\theta$ is the direction of the vector.

Formulas

The magnitude of a vector in a three-dimensional plane

Consider, a vector $\overrightarrow{x } = a_x\overrightarrow{i} + a_y\overrightarrow{j} + a_z\overrightarrow{k}$. Hence, the magnitude of the vector is calculated using the following formula:

$\left|\overrightarrow{x}\right| = \sqrt{{a^2}_x+{a^2}_y+{a_z}^2}$

Context and Applications

The magnitude is used in day-to-day life by the vehicles traveling by the means of air or water. The navigation of such vehicles is done using magnitude and vectors as the direction and magnitude both need to be provided. The magnitude is useful for students studying in schools, various undergraduate and postgraduate courses such as Bachelors in Engineering (Mechanical and Civil), Masters in Engineering (Mechanical and Civil), Bachelors in Science (Mathematics and Physics), and Masters in Science (Mathematics and Physics).

Practice Problems

Question 1) Which of the following represents the distance between two points in space?

a) magnitude of a vector

b) square of a vector

c) ending point of a vector

d) origin point of a vector

Answer – a)

Explanation – The magnitude of a vector represents the distance between the two points in space. The rest given options do not mean the distance between the two points in space.

Question 2) Which of the following is an element with both direction and magnitude?

a) scalar

b) radian

c) vector

d) units

Answer – c)

Explanation – A vector is an element with both direction and magnitude. Scalar is an element with only magnitude. Radian and unit are also not elements with both direction and magnitude.

Question 3) What is the set of vectors in space known as?

a) vector space

b) vector set

c) scalar space

d) scalar set

Answer – a)

Explanation – The set of vectors in space is known as the vector space. The rest given options do not mean the set of vectors in space.

Question 4) Which of the following is a vector that has a magnitude equal to zero?

a) position vector

b) parallel vector

c) unit vector

d) zero vector

Answer – d)

Explanation – A zero vector is a vector that has a magnitude equal to zero. A position vector is a vector that represents the direction and position of a vector in 3D space. Parallel vectors are vectors that have the same directions. Unit vectors are vectors that have a magnitude equal to 1.

Question 5) Which of the following is also known as a location vector?

a) parallel vector

b) negative vector

c) position vector

d) co-initial vector

Answer – c)

Explanation – A position vector is also known as a location vector. The rest given options of vectors are not known as location vectors.