Magnitude of a Vector Formula

Magnitude Formula

Do you want to know how to calculate magnitude? The magnitude formula of a vector can be found out in two aspects. In one instance, the magnitude is computed for a vector when its endpoint is at origin (0,0) while in the other instance, the beginning and ending point of the vector is at definite points (x1, y1) and (x2, y2) respectively. The Formula of Magnitude of a Vector to compute the length for each of the cases is as follows.


Magnitude of a Vector Formula


Magnitude Formula for a Vector when start points are (x1, y1) and endpoints are (x2, y2)

\[|v|=\sqrt{(x_{2}+x_{1})^{2}+(y_{2}+y_{1})^{2}}\] 

Magnitude Formula for a Vector When Ending Point is Origin

\[|v|=\sqrt{x^{2}+y^{2}}\]

 


Vectors in 2D

A vector with starting point at the origin and an ending point at (a, b) is written <a, b>. Geometrically, a vector is a vector implicated to a directed line segment, while algebraically it is referred to as an ordered pair. A vector can also be 3-dimensional.


Solved Examples Using The Magnitude Math Formula

Example:

Calculate the magnitude of the vector with \[\overrightarrow{u}\]  = (4,6) ?

Solution: 

Given, \[\overrightarrow{u}\] = (4,6)

Use Magnitude Formula,

\[|v|=\sqrt{x^{2}+y^{2}}\]

\[|v|=\sqrt{4^{2}+6^{2}}\]

\[|v|=\sqrt{16+36}\]

|v|= 7.22

 

Example:

Calculate the direction of the vector \[\overrightarrow{AB}\] whose starting point A is at (2,3)(2,3) and the terminal point at B is at (5,8).

The coordinates of the starting point and the ending point are already given.

Substitute them in the formula tanθ = y2 − y1 /x2 − x1

tanθ = 8 − 3/5 − 2            

=5/3

Calculate the inverse tan,

θ = tan−1(5/3)

≈59°

The vector \[\overrightarrow{AB}\] has a direction of about 59°

FAQs (Frequently Asked Questions)

1. What is the Magnitude of a Vector?

Answer: As we are familiar with, the vector is an object that consists of both the magnitude as well as direction. In order to identify the magnitude of a vector, we are required to find the length of the vector. Quantities such as force, velocity, momentum, displacement, etc. are vector quantities. But volume, mass, speed, distance, temperature, etc. are scalar quantities. The scalar consists of the only magnitude, for as much as the vectors have both magnitude and direction.


The magnitude of a vector formula is incorporated to find the length for a given vector (say v) and is represented as |v|. Thus typically, this quantity is the length between the starting point and endpoint of the vector. In order to compute the magnitude of the vector, we use the distance formula.

2. What is the Direction of a Vector?

Answer: The direction of a vector is merely the measurement of the angle which is formed with the horizontal line. One of the techniques to calculate the direction of the vector AB→ is;

Tan α = y/x; endpoint at 0.

Where,

x = the change in horizontal line

y = is the change in a vertical line.

Or tan α = y1–y0/x1–x0 ; where (x0 , y0) is the starting point and (x1 , y1) is the endpoint.