how to find the magnitude of a vector

Magnitude of a Vector Formula

The magnitude of a vector formula helps to summarize the numeric value for a given vector. A vector has a direction and a magnitude. The individual measures of the vector along the x-axis, y-axis, and z-axis are summarized using this magnitude of a vector formula. It is denoted by \(|\overrightarrow{ \text v}|\). The magnitude of a vector is always a positive number or zero, i.e., it cannot be a negative number. Let us understand the magnitude of a vector formula using a few solved examples in the end.

What Is the Magnitude of a Vector Formula?

The magnitude of a vector \(|\overrightarrow{ \text A}|\) is |A|. For a given vector with direction ratios along the x-axis, y-axis, and z-axis, the magnitude of the vector is equal to the square root of the sum of the square of its direction ratios. This can be clearly understood from the below magnitude of a vector formula.

Magnitude of a Vector Formula

For a vector \(\bar A = x \hat i + y \hat j + z \hat k \) its magnitude is:

\( |A| = \sqrt{x_1^2 + y_1^2 + z_1^2}\)

The magnitude is calculated for a vector when its endpoint is at origin (0,0).

\(|\overrightarrow{ \text v}| = \sqrt{x^2 + y^2}\)

The starting and ending point of the vector is at certain points (x₁, y₁) and (x₂, y₂) respectively.

\(|\overrightarrow{ \text v}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

To determine the magnitude of a two-dimensional vector from its coordinates,

Step 1: Identify its components.
Step 2: Find the sum of the square of each of its components.
Step 3: Take the square root of the sum so obtained.

Thus, the formula to determine the magnitude of a vector v = (x\(_1\), y\(_1\)) is: \(|\overrightarrow{ \text v}| = \sqrt{x^2 + y^2}\). This formula is derived from the Pythagorean theorem.

To determine the magnitude of a three-dimensional vector from its coordinates,

Step 1: Identify its components.
Step 2: Find the sum of the square of each of its components.
Step 3: Take the square root of the sum so obtained.

Thus, the formula to determine the magnitude of a vectorV = (x\(_1\), y\(_1\), z\(_1\)) is: |V| = \(\sqrt{x_1^2 + y_1^2 + z_1^2}\)

Let us see the applications of the magnitude formula in the following section.

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Examples Using Magnitude of a Vector Formula

Example 1: Using magnitude formula, find the magnitude of the vector with \(\overrightarrow{\text u}\) = (2, 5) ?

Solution:

To find: Magnitude of the given vector

Given:

Vector \(\overrightarrow{ \text u}\) = (2,5)

Using magnitude formula,

\(|\overrightarrow{ \text u}| \)= √(x² + y²)

= √(2² + 5²)

= √(4 + 25)

\(|\overrightarrow{ \text u}| \) = 5.385

Answer: Magnitude of the given vector = 5.385

Example 2: Find the magnitude of the vector \(3\hat i + 4\hat j - 5\hat k\).

Solution:

To find: Magnitude of the given vector

Given vector \(\bar A = 3\hat i + 4\hat j - 5\hat k\)
|A| = √(3² + 4² + (-5)²) =

√(9 + 16 + 25) =√50

=5√2

Answer: Magnitude of the given vector = 5√2

Example 3:Find the magnitude of the vector \(5\hat i - 4\hat j + 2\hat k\).

Solution:

To find: Magnitude of the given vector

Given vector \(\bar A = 5\hat i - 4\hat j + 2\hat k\)
|A| =√(5² + (-4)² + 2²)

√(25 + 16 + 4) = √45

=3√ 5

Answer: Magnitude of the given vector = 3√5

FAQs on Magnitude of a Vector Formula

What Is the Magnitude of a Vector Formula?

The magnitude of a vector formula summarizes the numeric value for a given vector. It is denoted by \(|\overrightarrow{ \text v}|\). The magnitude of vector formulas are as follows:

\(|A| = \sqrt{x_1^2 + y_1^2 + z_1^2}\) for a vector \(\bar A = x \hat i + y \hat j + z \hat k \) its magnitude is:
\(|\overrightarrow{ \text v}| = \sqrt{x^2 + y^2}\) when its endpoint is at origin (0,0)
\(|\overrightarrow{ \text v}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) when the starting and ending point of the vector is at certain points (x₁, y₁) and (x₂, y₂)

How To Use the Magnitude of a Vector Formula?

In order to use the magnitude of a vector formula, follow the steps given below

Step 1: Check for the given parameters.
Step 2: Put the values in the appropriate formula

For a vector \(\bar A = x \hat i + y \hat j + z \hat k \) its magnitude is \(|A| = \sqrt{x_1^2 + y_1^2 + z_1^2}\)

The magnitude of a vector when its endpoint is at origin (0,0) then \(|\overrightarrow{ \text v}| = \sqrt{x^2 + y^2}\)

The starting and ending point of the vector is at certain points (x₁, y₁) and (x₂, y₂) then \(|\overrightarrow{ \text v}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

What Concept Is Behind the Formula For Calculating the Magnitude of a Vector?

The magnitude of a vector refers to the length or size of the vector. It also determines its direction. The concepts behind these formulas include the Pythagorean theorem and the distance formula, which are used to derive the formula of the magnitude of the vector.

What Is the Magnitude of Vector Formula In Words?

For a given vector with direction ratios along the x-axis, y-axis, and z-axis, the magnitude of the vector is equal to the square root of the sum of the square of its direction ratios.