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: Worksheet 1

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: \[\begin{vmatrix} \vec{h} \end{vmatrix} = 5\]

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