Magnitude of a Vector

Magnitude of a Vector

The magnitude of a vector (or modulus of a vector) is a scalar quantity that represents its length or size. It tells you “how much” of a quantity the vector represents, without regard to its direction.

Think of a vector as an arrow pointing from a starting point to an ending point. The magnitude is the length of that arrow

The following diagram shows how to find the magnitude of a 2-D vector. Scroll down the page for more examples and solutions to calculate the magnitudes of 2-D and 3-D vectors.

 

The following diagram shows how to find the magnitude of a 3-D Vector.

 

How to Calculate the Magnitude of a Vector
The formula for the magnitude of a vector is derived from the Pythagorean theorem, as a vector’s components can be thought of as the legs of a right triangle, and the magnitude is its hypotenuse.

1. In Two Dimensions (2D)
   If a vector v has components (x,y), its magnitude is calculated as:
   \(\left| v \right|=\sqrt{x^{2}+y^{2}}\)
   
   Example: Find the magnitude of vector \textbf{v}=\left( 3,4 \right)
   \(\left| v \right|=\sqrt{3^{2}+4^{2}}=5\)

2. In Three Dimensions (3D)
   If a vector v has components (x,y,z), its magnitude is calculated as:
   \(\left| v \right|=\sqrt{x^{2}+y^{2}}+z^{2}\)
   
   Example: Find the magnitude of vector \(\textbf{w}=\left( 2,-3,6 \right)\)
   \(\left| w \right|=\sqrt{2^{2}+(-3)^{2}+6^{2}}=7\)
   

Example (2D vector):
Express each of the following vectors as a column vector and find its magnitude.

 

Key characteristics of magnitude:

1. It is always a non-negative value (either positive or zero). A magnitude of zero means the vector is a “zero vector” (a point).
2. It is a scalar quantity, meaning it has no direction.
3. It is often denoted by double vertical bars around the vector symbol, like ||v|| or single vertical bars, |v|.

Videos

Vectors in 2D

Adding vectors geometrically, scalar multiplication, how to find the magnitude and direction angle of a vector.
A vector with initial point at the origin and terminal point at (a, b) is written <a, b>.
Geometrically, a vector is a directed line segment, while algebraically it is an ordered pair.

Example:
Find the magnitude and the direction angle for u = <-3, 4>

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Vectors: magnitude of a vector in 2D.

Example:
Find the magnitude of the following vectors:
a = 4i - 3j
b = -2i + 5j

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Vectors in 3D

A vector can also be 3-dimensional.
The following video gives the formula, and some examples of finding the magnitude, or length, of a 3-dimensional vector.

Example:
Find the magnitude:
a = <3, 1, -2>
b = 5i -j + 2k

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Vectors : Magnitude of a vector 3D.

Examples:

1. Find the magnitude of a = 4i + 3j + 2k
2. If A(3, -5, 6) and B(4, 1, 3) find the length AB.

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