# Magnitude of a Vector - How to Calculate It

## (also called Modulus of a Vector)

Every vector has a magnitude, also called modulus. In this section we learn how to calculate the magnitude of a vector.

### Magnitude (Modulus) of a Vector

Given a vector $$\vec{v} = \begin{pmatrix}x \\ y \end{pmatrix}$$, its magnitude, also called modulus can be calculated with the formula: $\begin{vmatrix}\vec{v} \end{vmatrix} = \sqrt{x^2 + y^2}$

### Tutorial: Magnitude of a Vector

In this tutorial we learn how to calculate the magnitude of a vector.

### Example 1

Given the two vector $$\vec{a} = \begin{pmatrix}1 \\ 3 \end{pmatrix}$$ we can calculate its magnitude as follows: \begin{aligned} \begin{vmatrix} \vec{a} \end{vmatrix} & = \sqrt{1^2 + 3^2} \\ & = \sqrt{1 + 9} \\ \begin{vmatrix} \vec{a} \end{vmatrix} & = \sqrt{10} \quad \begin{pmatrix} \approx 3.16 \end{pmatrix} \end{aligned}

## Exercise 1

Find and write the exact value of the magnitude for each of the vectors, shown below. When applicable: also use your calculator to round your answers to two decimal places (2 d.p).

1. For vector $$\vec{a} = \begin{pmatrix} 8 \\ 6 \end{pmatrix}$$.
2. For vector $$\vec{b} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}$$.
3. For vector $$\vec{c} = \begin{pmatrix} -2 \\ 3 \end{pmatrix}$$.
4. For vector $$\vec{d} = \begin{pmatrix} -4 \\ 5 \end{pmatrix}$$.
1. For vector $$\vec{e} = \begin{pmatrix} 0 \\ 4 \end{pmatrix}$$.
2. For vector $$\vec{f} = \begin{pmatrix} 2 \\ -2 \end{pmatrix}$$.
3. For vector $$\vec{g} = \begin{pmatrix} 5 \\ -12 \end{pmatrix}$$.
4. For vector $$\vec{h} = \begin{pmatrix} -4 \\ 3 \end{pmatrix}$$.
Note: this exercise can be downloaded as a worksheet to practice with:

## Solution Without Working

1. For vector $$\vec{a} = \begin{pmatrix} 8 \\ 6 \end{pmatrix}$$, we find: $\begin{vmatrix} \vec{a} \end{vmatrix} = 10$

2. For vector $$\vec{b} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}$$, we find: $\begin{vmatrix} \vec{b} \end{vmatrix} = \sqrt{29} \quad \begin{pmatrix}5.39 \ \text{2 d.p} \end{pmatrix}$

3. For vector $$\vec{c} = \begin{pmatrix} -2 \\ 3 \end{pmatrix}$$, we find: $\begin{vmatrix} \vec{c} \end{vmatrix} = \sqrt{13} \quad \begin{pmatrix}3.61 \ \text{2 d.p} \end{pmatrix}$

4. For vector $$\vec{d} = \begin{pmatrix} -4 \\ 5 \end{pmatrix}$$, we find: $\begin{vmatrix} \vec{d} \end{vmatrix} = \sqrt{41} \quad \begin{pmatrix}6.40 \ \text{2 d.p} \end{pmatrix}$
1. For vector $$\vec{e} = \begin{pmatrix} 0 \\ 4 \end{pmatrix}$$, we find: $\begin{vmatrix} \vec{e} \end{vmatrix} = 4$

2. For vector $$\vec{f} = \begin{pmatrix} 2 \\ -2 \end{pmatrix}$$, we find: $\begin{vmatrix} \vec{f} \end{vmatrix} = \sqrt{8} = 2\sqrt{2} \quad \begin{pmatrix}2.83 \ \text{2 d.p} \end{pmatrix}$

3. For vector $$\vec{g} = \begin{pmatrix} 5 \\ -12 \end{pmatrix}$$, we find: $\begin{vmatrix} \vec{g} \end{vmatrix} = 13$

4. For vector $$\vec{h} = \begin{pmatrix} -4 \\ 3 \end{pmatrix}$$, we find: $\begin{vmatrix} \vec{h} \end{vmatrix} = 5$