Magnitude of a Vector

Vector quantities are the quantities that have both direction and magnitude. The magnitude of a vector is the length of the vector. It is given by the numeric value of the vector and as it represents the length of the vector so it is always positive. For any vector [Tex]\overrightarrow{A}  [/Tex] its magnitude is represented as [Tex]\overrightarrow{A} [/Tex].

Let’s learn more about the magnitude of the vector its formula, examples, and other in this article.

What is the Magnitude of a Vector?

Magnitude of a vector is defined as the length of the vector. As the magnitude of the vector denotes the length of the vector it is always positive. For any vector A its magnitude is represented as |A|. Suppose a vector is defined as xi + yj then its magnitude is defined as the square root of the sum of squares of the individual terms. The magnitude of the vector represents the length of the vector i.e. the value or impact the vector has. 

For example, if a force of 5i N works on an object then its magnitude is 5 N which signifies that the strength of the force applied is 5 N, and ‘i’ in 5i represents that it is applied in the positive x direction.

Magnitude of a Vector Formula

There are different ways to calculate the magnitude of the vector. Based on the given data, use a different kind of formula to find the magnitude of a vector. The magnitude of a vector A is represented using the modulus operator i.e., |A|

There are various formulas that are used to count the magnitude of the vector. The following image shows the important formulas used to find the magnitude of the vector.

Following are the ways to calculate the magnitude. 

• If they are given a vector Ā = xi+ yĵ + zk̂  then the magnitude of vector Ā can be calculated using the below formula

Magnitude of vector Ā (|A|) = √(x² + y² +z²)

• If the starting point vector is say (x₁, y₁) and endpoint of a vector is say (x₂, y₂)  are given then the magnitude of the vector [Tex]\overrightarrow{AB}      [/Tex] is given by,

The magnitude of a vector, when the start and endpoints of a vector are given, is nothing but the distance between the points. The formula for finding magnitude is given by

[Tex]\overrightarrow{AB}  [/Tex]     = [Tex]\sqrt{((x_2-x_1)^2+(y_2-y_1)^2)}[/Tex]

• If any of the starting or endpoint of a vector is at origin o(0, 0) and another point is A(x, y) like specified in the below figure,

Then the formula for finding the magnitude of a vector where one of the ends of a vector is at the origin is given by

|Ā| = √(x²+y²)

Direction of a Vector

Vector quantities are quantities that have both magnitudes and directions. The direction of the vector quantity signifies in which direction the vector quantity is applied. It is defined as the angle which the vector makes with the horizontal line or the x-axis. It is represented by the symbol α.

The image below shows the arrow which is used to show the direction of the vector.

It is calculated using the formula,

α = tan^-1(y/x)

For the vector generated by the coordinates (x₁, y₁) and (x₂, y₂) their direction is given by the formula,

α = tan^-1[(y₂ – y₁)/(x₂ – x₁)]

How to Find Magnitude of a Vector?

The magnitude of the vector is calculated using the steps discussed below,

Step 1: Identify the x, y, and z components of the vector.

Step 2: Find the square of all the x, y, and z components.

Step 3: Add all the squares found in Step 2.

Step 4: Find the square root of the sum obtained in Step 3. 

The value obtained after step 4 is the magnitude of the given vector.

Example: Find the magnitude of the vector A = 3i + 4j

Solution:

The magnitude of vector A is calculated using the steps discussed above.

Step 1: Comparing A = 3i + 4j with xi + yj we get x = 3 and y = 4

Step 2: x² = 3²  = 9 and y² = 4² = 16

Step 3: x² + y² = 9 + 16 = 25

Step 4: √(25) = 5

Thus, the magnitude of the vector  A = 3i + 4j is 5 units.

Read More,

• Scalar and Vector
• Vector Operations
• How to calculate the Unit Vector?

Solved Examples on Magnitude of Vector

Example 1: Find the magnitude for the vector Ā = 2i + 3ĵ + 4k.

Solution:

Given,

Ā = 2i + 3ĵ + 4k

Magnitude |A| = [Tex]\sqrt{(2^2+3^2+4^2)}[/Tex]

                       = [Tex]\sqrt{(4+9+16)}      [/Tex]
                       = √29
                       = 5.38

The magnitude of vector 2i+ 3ĵ + 4k is 5.38 unit

Example 2: Find the magnitude for the vector Ā = 3i + 3ĵ – 6k

Solution:

Given

Ā = 3i + 3ĵ – 6k

Magnitude |A| = [Tex]\sqrt{(3^2+3^2+(-6)^2)}[/Tex]

                       = [Tex]\sqrt{(9+9+36)}      [/Tex]
                       = √54
                       = 7.35

The magnitude of vector 3i+ 3ĵ – 6k is 7.35 unit.

Example 3: Find the magnitude of the vector if the starting point of a vector is (3, 4) and the ending point is (6, 2).

Solution:

Given,

(x₁, y₁) = (3, 4)
(x₂, y₂) = (6, 2)

|Ā|= [Tex]\sqrt{((x_2-x_1)^2+(y_2-y_1)^2)}[/Tex]

    = [Tex]\sqrt{((6-3)^2+(2-4)^2)}      [/Tex]
    = √(3² + (-2)²)
    = √(9+4)
    = √13 = 3.6

Thus, the magnitude of the given vector is 3.6 unit.

Example 4: Find the magnitude of the vector if the starting point of a vector is (2, 1, 4) and the ending point is (5, 2, 6).

Solution:

Given,

(x₁, y₁, z₁) = (2, 1, 4)

(x₂, y₂, z₂) = (5, 2, 6)

|Ā| = [Tex]\sqrt{((x_2-x_1)^2+(y_2-y_1)^2)}[/Tex]

     = [Tex]\sqrt{((5-2)^2+(2-1)^2+(6-4)^2)}      [/Tex]
     = [Tex]\sqrt{(32+12+22)}      [/Tex]
     = √(9 +1 + 4) 
     = √14 = 3.74

Thus, the magnitude of the given vector is 3.74 unit.

Example 5: What is the magnitude of the vector which starts at the origin and endpoint at (3, 4).

Solution:

Given,

Starting Point of vector is O(0, 0)

End Point (x, y) = (3, 4)

Magnitude of Vector (|Ā|) = √(x²+y²)

                                        = √(3² + 4²)
                                        = √(9 + 16)
                                        = √25 = 5

Thus, the magnitude of the given vector is 5 unit.

Example 6: Find the magnitude of the vector in which one of the endpoints is at the origin and the other point at (1, 4, 3).

Solution:

Given,

End Point of vector is O(0, 0)

Other Point (x, y, z) = (1, 4, 3)

Magnitude of Vector (|Ā|) = √(x²+y²+z²)

                                        = [Tex]\sqrt{(1^2+4^2+3^2)}      [/Tex]
                                        = [Tex]\sqrt{(1+16+9)}      [/Tex]
                                        = √26 = 5.09

Thus, the magnitude of the given vector is 5.09 unit.

FAQs on Magnitude of a Vector

Q1: What is the Magnitude of a Vector Formula?

Answer:

The magnitude of a vector is the numeric value of the vector and it defines the length of the vector. For any vector, A its magnitude is represented as |A|. The magnitude of the vector is calculated using the formula,

For any vector, A = xi + yj + zk its magnitude is given by the formula 

|A| = √(x² + y² + z²)

For any vector whose starting point and end point are respectively (x₁, y₁) and (x₂, y₂)  its magnitude is given by the formula 

|A| = √((x₂ – x₁)² + (y₂ – y₁)²)

Q2: How to represent the Magnitude of a Vector?

Answer:

The magnitude of the vector A is represented by the symbol |A|.

Q3: How to find the Magnitude of a Vector?

Answer:

Various formulas are used to calculate the magnitude of the vector some of them are,

• |A| = √(x² + y² + z²) when the vector is in the form of A = xi + yj + zk
• |A| = √((x)² + (y)²) when the vector is given by point A (x, y) and the origin O(0, 0).
• |A| = √((x₂ – x₁)² + (y₂ – y₁)²) when the vector is given by point A (x₁, y₂) and point B (x₂, y₂).

Q4: Find a vector of Magnitude 5.

Answer:

There are various vectors which can have a magnitude of 5 an example of which is vector A represented as,

A = 3i + 4j Or A = 4i + 5j