How to find the magnitude of a vector

A vector is a quantity that has a magnitude and a direction. In various areas of physics and mathematics, vectors are used to understand the behavior of directional quantities in two and three-dimensional spaces.

In this Answer, we will discuss how we can find the magnitude of a vector.

The magnitude of a vector

The magnitude of a vector, also known as length or norm, is defined as a scalarA numerical value. that represents its length or strength. Suppose we have a vector $\vec{v}$, we represent its magnitude as $|v|$.

The following diagram illustrates the magnitude part of a vector.

Calculation of a vector’s magnitude

To find the magnitude of a vector, we have two kinds of information available: the vector with its components or the vector with its endpoints. Let's discuss how to find the magnitude in both cases.

1) When a vector's components are given

Let's first understand what a vector with its component means using the following diagram:

To find the magnitude of a vector with its components' values, we follow the following steps:

• We identify its components.

• We square each component and add these squared values.

• We take the square root of the sum calculated in the above step.

Let's use the above steps to find a mathematical formula of the vector's magnitude in two and three-dimensional space. Suppose we have a vector $\vec{v}$.

In a 2D space, we can represent its magnitude $|v|$ as:

• $x$ the magnitude along the x-axis

• $y$ is the magnitude along the y-axis

In a 3D space, we can represent its magnitude $|v|$ as:

• $x$ the magnitude along the x-axis

• $y$ is the magnitude along the y-axis

• $z$ is the magnitude along the z-axis

Example

Suppose we have a vector $\vec{v}$ such that:
$\vec{v} = (3,4)$

Let's calculate its magnitude $|v|$:

2) When a vector's endpoints are given

Let's first understand what a vector with its endpoints means using the following diagram:

To find the magnitude of a vector with its endpointsStarting and ending point. values, we follow the following steps:

• We subtract the starting point's coordinates from the ending point's coordinates.

• We square each of these differences and add these squared values.

• We take the square root of the sum calculated in the above step.

Let's use the above steps to find a mathematical formula of the vector's magnitude in two and three-dimensional space. Suppose we have a vector $\vec{v}$.

Suppose in a 2D space, we have two points $P1$ and $P2$ such that:

$P1=(x1,y1)$ (Starting point of the vector)

$P2=(x2,y2)$ (Ending point of the vector)

Let's represent its magnitude $|v|$ as:

Suppose in a 3D space, we have two points $P1$ and $P2$ such that:

$P1=(x1,y1,z1)$ (Starting point of the vector)

$P2=(x2,y2,z2)$ (Ending point of the vector)

Let's represent its magnitude $|v|$ as:

Note: When either the starting or endpoint of a vector is at the originAll coordinates have values zero. and the other point is $v1(x1,y1)$, then the formula becomes:

$|v|=\sqrt{x1^2 + y1^2}$

Example

Suppose we have a vector $v$ with two endpoints $P1$ and $P2$ such that:

$P1 = (1,2)$ (Starting point of the vector)

$P2 = (3,4)$ (Ending point of the vector)

Let's calculate its magnitude $|v|$:

Conclusion

We can find the magnitude of a vector after following a series of steps based on the vector's data provided. This magnitude helps us determine a vector's length, disregarding its direction. Moreover, it is extremely helpful in physics, engineering, and computer graphics to understand the movement of objects, compare their strengths, and make predictions.