Home / IGCSE Maths 0580 / Transformations and vectors / Magnitude of a vector
Cambridge IGCSE · Topic 7.3
IGCSE Maths 0580 / Topic 7 / IGCSE

How long is the arrow?

Extended. The magnitude of a vector is its length. The column vector gives the two shorter sides of a right-angled triangle, so Pythagoras does the rest.

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The key idea

The magnitude of \(\begin{pmatrix}x\\y\end{pmatrix}\) is \(\sqrt{x^2 + y^2}\). It is written \(|\mathbf{a}|\) or \(|\overrightarrow{AB}|\), it is always positive, and \(|\mathbf{a} + \mathbf{b}|\) is usually NOT \(|\mathbf{a}| + |\mathbf{b}|\).

4 3 v v = (4, 3) |v| = √(4² + 3²) = √25 = 5
Fig. 1: v = (4, 3) is the hypotenuse of a right-angled triangle with sides 4 and 3, so |v| = 5
Section 01

Pythagoras for vectors.

In the lab, drag the head of the vector and watch the right-angled triangle beneath it. Read the two components, square them, add, square-root, and compare with the magnitude the lab shows. Try a vector with a negative component: does the length change sign?

Worked example A whole-number magnitude
Find \(|\mathbf{a}|\) when \(\mathbf{a} = \begin{pmatrix}3\\-4\end{pmatrix}\).
  1. \(|\mathbf{a}| = \sqrt{3^2 + (-4)^2}\).
  2. \(= \sqrt{9 + 16} = \sqrt{25}\).
Answer: \(|\mathbf{a}| = 5\)
Worked example Between two points
\(A\) is \((2, -1)\) and \(B\) is \((-4, 7)\). Find \(|\overrightarrow{AB}|\).
  1. \(\overrightarrow{AB} = \begin{pmatrix}-4 - 2\\7 - (-1)\end{pmatrix} = \begin{pmatrix}-6\\8\end{pmatrix}\).
  2. \(|\overrightarrow{AB}| = \sqrt{36 + 64} = \sqrt{100}\).
Answer: \(|\overrightarrow{AB}| = 10\)
Worked example Rounding
Find the magnitude of \(\begin{pmatrix}5\\2\end{pmatrix}\), correct to 3 significant figures.
  1. \(\sqrt{5^2 + 2^2} = \sqrt{29}\).
  2. \(\sqrt{29} = 5.385\ldots\)
Answer: \(5.39\) (3 s.f.)
Examiner trap

Squaring a negative number on a calculator without brackets: typing \(-4^2\) gives \(-16\). Type \((-4)^2\), or simply square the number without its sign, since the square is always positive.

Section 02

Magnitude of a combination.

To find the magnitude of something like \(2\mathbf{a} - \mathbf{b}\), first work out the column vector, then use Pythagoras once on the result.

Worked example Sum first, then length
\(\mathbf{a} = \begin{pmatrix}3\\1\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}1\\2\end{pmatrix}\). Find \(|\mathbf{a} + \mathbf{b}|\) and compare it with \(|\mathbf{a}| + |\mathbf{b}|\).
  1. \(\mathbf{a} + \mathbf{b} = \begin{pmatrix}4\\3\end{pmatrix}\), so \(|\mathbf{a} + \mathbf{b}| = \sqrt{16 + 9} = 5\).
  2. \(|\mathbf{a}| = \sqrt{10} = 3.162\ldots\) and \(|\mathbf{b}| = \sqrt{5} = 2.236\ldots\)
  3. \(|\mathbf{a}| + |\mathbf{b}| = 5.40\) (3 s.f.), which is longer: going round two sides of a triangle is longer than the direct route.
Answer: \(|\mathbf{a} + \mathbf{b}| = 5\), but \(|\mathbf{a}| + |\mathbf{b}| = 5.40\)
Worked example A scalar combination
With the same \(\mathbf{a}\) and \(\mathbf{b}\), find \(|2\mathbf{a} - \mathbf{b}|\).
  1. \(2\mathbf{a} - \mathbf{b} = \begin{pmatrix}6 - 1\\2 - 2\end{pmatrix} = \begin{pmatrix}5\\0\end{pmatrix}\).
  2. \(\sqrt{5^2 + 0^2} = 5\).
Answer: \(|2\mathbf{a} - \mathbf{b}| = 5\)
Your turn
\(\mathbf{p} = \begin{pmatrix}-2\\5\end{pmatrix}\). Find \(|3\mathbf{p}|\), correct to 3 significant figures.
Examiner trap

Adding magnitudes. \(|\mathbf{a} + \mathbf{b}|\) is the length of ONE vector; find the column vector \(\mathbf{a} + \mathbf{b}\) first, then its length. \(|\mathbf{a}| + |\mathbf{b}|\) is a different number.

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What is the magnitude of \(\begin{pmatrix}6\\8\end{pmatrix}\)?

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What is the magnitude of \(\begin{pmatrix}-5\\12\end{pmatrix}\)?

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What does \(|\overrightarrow{AB}|\) mean?

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What is the exact magnitude of \(\begin{pmatrix}2\\-3\end{pmatrix}\)?

Section 03

Unknowns and speeds.

If the magnitude is given, Pythagoras becomes an equation for a missing component. And when a vector describes a movement per hour, its magnitude is a speed.

Worked example A missing component
\(\left|\begin{pmatrix}k\\12\end{pmatrix}\right| = 13\). Find the possible values of \(k\).
  1. \(k^2 + 12^2 = 13^2\), so \(k^2 = 169 - 144 = 25\).
  2. \(k = 5\) or \(k = -5\).
Answer: \(k = \pm 5\)
Worked example A quadratic in k
\(\left|\begin{pmatrix}k\\k + 1\end{pmatrix}\right| = 5\). Find the possible values of \(k\).
  1. \(k^2 + (k + 1)^2 = 25\), so \(k^2 + k^2 + 2k + 1 = 25\).
  2. \(2k^2 + 2k - 24 = 0\), so \(k^2 + k - 12 = 0\).
  3. \((k + 4)(k - 3) = 0\).
Answer: \(k = 3\) or \(k = -4\)
Worked example Velocity to speed
A ship moves by \(\begin{pmatrix}6\\8\end{pmatrix}\) km every hour. Find its speed.
  1. Speed is the distance covered each hour, which is the magnitude.
  2. \(\sqrt{36 + 64} = \sqrt{100} = 10\).
Answer: 10 km/h
Examiner trap

Forgetting the negative root. \(k^2 = 25\) has two solutions, \(5\) and \(-5\): both \(\begin{pmatrix}5\\12\end{pmatrix}\) and \(\begin{pmatrix}-5\\12\end{pmatrix}\) have length 13.

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\(\mathbf{a} = \begin{pmatrix}4\\-1\end{pmatrix}\) and \(\mathbf{b} = \begin{pmatrix}-1\\5\end{pmatrix}\). Find \(|\mathbf{a} + \mathbf{b}|\).

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\(A\) is \((-3, 2)\) and \(B\) is \((5, -4)\). Find \(|\overrightarrow{AB}|\).

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\(\left|\begin{pmatrix}k\\12\end{pmatrix}\right| = 13\) and \(k > 0\). Find \(k\).

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A drone moves by \(\begin{pmatrix}-9\\12\end{pmatrix}\) km every hour. How far does it travel in 2.5 hours?

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Magnitudes, measured.

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Six original Cambridge-style questions
Magnitudes of column vectors and of combinations, exact surds, a missing component and a drone's flight.