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# Magnitude of a Vector

The magnitude of a vector formula helps to summarize the numeric value for a given vector. The magnitude of a vector **v** whose components are given by <x_{1}, y_{1}> is given by the formula |**v**| =√(x_{1}^{2} + y_{1}^{2}).

A vector has a direction and a magnitude. The individual measures of the vector **v** along the x-axis, y-axis, and z-axis are encapsulated using this magnitude of a vector formula. It is denoted by |**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?

The **magnitude of a vector A** is the **length of the vector A **and is denoted by |**A**|. It is the square root of the sum of squares of the components of the vector. 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 squares 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
**A**= x_{1}**i**+ y_{1}**j**+ z_{1}**k**, its magnitude is: |A| = √(x_{1}^{2}+ y_{1}^{2}+ z_{1}^{2}) - For a vector
**v**when one of its endpoints is at origin (0,0) and the other endpoint is at (x, y), its magnitude is: |**v**| = √(x^{2}+ y^{2}) - For a vector
**v**with endpoints at (x_{1}, y_{1}) and (x_{2}, y_{2}), its magnitude is: |**v**| = √((x_{2}- x_{1})^{2}+ (y_{2}- y_{1})^{2})

## How to Find Magnitude of a Vector?

To find magnitude of vector:

**Step 1**: Identify its components.**Step 2**: Find the sum of the squares 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 (in two-dimensional space)
**v**= (x, y) is: |**v**| =√(x^{2}+ y^{2}). This formula is derived from the Pythagorean theorem. - the formula to determine the magnitude of a vector (in three-dimensional space)
**V**= (x, y, z) is: |**V**| = √(x^{2}+ y^{2}+ z^{2})

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

## Examples Using Magnitude of a Vector Formula

**Example 1:** Using the magnitude formula, find the magnitude of the vector with **u** = (2, 5)?

**Solution:**

To find: Magnitude of the given vector

Given:

Vector **u** = (2,5)

Using magnitude formula,

|**u**| = √(x^{2} + y^{2})

= √(2^{2} + 5^{2})

= √(4 + 25)

= √29

≈ 5.385

**☛ Also Check:** How to find the magnitude of a vector with 3 components

**Answer: **Magnitude of the given vector = 5.385

**Example 2:** Find the magnitude of the vector 3**i** + 4**j** - 5**k**.

**Solution:**

To find: Magnitude of the given vector

Given vector **A** = 3**i** + 4**j** - 5**k**,

|A| = √(3^{2} + 4^{2} + (-5)^{2})

= √(9 + 16 + 25)

= √50

= 5√2

**Answer: **Magnitude of the given vector = 5√2

**Example 3: **Find the length of the vector whose endpoints are given by the position vectors <-3, 1, 7> and <2, -5, 9>.

**Solution:**

To find: Length of the given vector

The endpoints of the given vector are **OA** = <-3, -1, 7> = <x_{1}, y_{1}, z_{1}> and **OB** = <2, -5, 9> = <x_{2}, y_{2}, z_{2}>

Its magnitude is calculated by the vector formula:

|A| = √((x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}) + (z_{2} - z_{1})^{2}

= √ [(2 - (-3))^{2} + (-5-(-1))^{2} + (9 - 7)^{2}]

= √(5^{2} + (-4)^{2} + 2^{2})

= √(25 + 16 + 4)

= √45

= 3√5

**Answer:** Length of the 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 **v**. It is denoted by |**v**|. The magnitude of vector formulas are as follows:

- |
**A**| =√(x^{2}+ y^{2}+ z^{2}) for a vector**A**= x**i**+ y**j**+ z**k** - |
**v**| =√(x^{2}+ y^{2}) when its endpoints are at origin (0, 0) and (x, y). - |
**v**| =√((x_{2}- x_{1})^{2}+ (y_{2}- y_{1})^{2}) when the starting and ending point of the vector at certain points (x_{1}, y_{1}) and (x_{2}, y_{2}) respectively.

### How to Calculate Magnitude of a Vector?

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

**Step 1:**Check for the given parameters.**Step 2:**Put the values in the appropriate formula- For a vector
**A**= x**i**+ y**j**+ z**k**its magnitude is |A| =√(x^{2}+ y^{2}+ z^{2}) - The magnitude of a vector when its endpoint is at origin (0,0) then |
**v**| =√(x^{2}+ y^{2}) - The starting and ending point of the vector is at certain points (x
_{1}, y_{1}) and (x_{2}, y_{2}) then |**v**| =√((x_{2}- x_{1})^{2}+ (y_{2}- y_{1})^{2})

- For a vector

### What is the Magnitude of Cross Product?

The cross product of two vectors **a** and **b** is given by the formula, **a** × **b** = |**a**| |**b**| sin θ \(\hat{n}\). To find the magnitude of cross product, we take the magnitude sign on both sides here. Then we get |**a** × **b|** = |**a**| |**b**| sin θ |\(\hat{n}\)|. Here, \(\hat{n}\) is a unit vector and hence its magnitude is 1. Therefore, the magnitude of cross product of vectors is |**a** × **b|** = |**a**| |**b**| sin θ.

### 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 Unit Vector?

A unit vector is a vector whose magnitude is 1. Hence, the magnitude of unit vector is 1.

### 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 squares of its direction ratios.

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