Magnitude of a vector

What the magnitude of a vector is Extended

The magnitude of a vector is its size: the length of the arrow that represents it. It is written with modulus signs:

• $\left|\mathbf{a}\right|$ is the magnitude of $\mathbf{a}$;
• $\left|\overrightarrow{AB}\right|$ is the magnitude of $\overrightarrow{AB}$, which is the distance from A to B.

For a column vector the magnitude comes from Pythagoras. The vector $\left(\begin{array}{c}x\\ y\end{array}\right)$ is the hypotenuse of a right-angled triangle whose other sides are $x$ across and $y$ up, so
$$\left|\left(\begin{array}{c}x\\ y\end{array}\right)\right|=\sqrt{x^2+y^2}$$

For example, $\left|\left(\begin{array}{c}4\\ 3\end{array}\right)\right|=\sqrt{16+9}=\sqrt{25}=5$.

A negative component makes no difference to the length: $(-3{)}^2=9$, the same as $3^2$. The vectors $\left(\begin{array}{c}4\\ 3\end{array}\right)$, $\left(\begin{array}{c}-4\\ 3\end{array}\right)$ and $\left(\begin{array}{c}4\\ -3\end{array}\right)$ all have magnitude 5; they just point different ways. A magnitude is a length, so it is never negative.

The vector $\left(\begin{array}{c}4\\ 3\end{array}\right)$ is the hypotenuse of a right-angled triangle with sides 4 and 3, so its magnitude is $\sqrt{4^2+3^2}=5$.

Exact answers and rounded answers Extended

Many magnitudes are not whole numbers. How you give them depends on the paper and the question.

• Without a calculator (Paper 2), or when the question asks for an exact answer, leave the answer as a surd in its simplest form. Look for the largest square number that divides the number under the root: $\sqrt{45}=\sqrt{9\times 5}=3\sqrt{5}$.
• With a calculator (Paper 4), give the answer correct to 3 significant figures unless the question says otherwise: $\sqrt{45}=6.708\dots =6.71$.

Keep the full calculator value in your working until the very end. If you need a magnitude in a later step, use the exact value or the full display, never the rounded answer.

Remember the order in Pythagoras: square each component, add, then square root. A common slip is to stop before the square root, which gives the magnitude squared.

Magnitude of a combination of vectors Extended

To find the magnitude of something like $\mathbf{a}+\mathbf{b}$ or $2\mathbf{a}-3\mathbf{b}$:

1. Work out the combination as a single column vector.
2. Then find the magnitude of that vector.

You cannot add magnitudes: in general $|\mathbf{a}+\mathbf{b}|\ne \left|\mathbf{a}\right|+\left|\mathbf{b}\right|$. The arrows $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{a}+\mathbf{b}$ form a triangle, and one side of a triangle is shorter than the other two together. The two are equal only when $\mathbf{a}$ and $\mathbf{b}$ point the same way.

Multiplying by a number does scale the magnitude: $\left|k\mathbf{a}\right|=\left|k\right|\times \left|\mathbf{a}\right|$. So $\left|3\mathbf{a}\right|=3\left|\mathbf{a}\right|$ and $|-2\mathbf{a}|=2\left|\mathbf{a}\right|$. This lets you find a vector in a given direction with a given length: if $\left|\mathbf{a}\right|=10$, the vector in the direction of $\mathbf{a}$ with magnitude 25 is $2.5\mathbf{a}$.

Distances and unknown components Extended

The distance between two points A and B is $\left|\overrightarrow{AB}\right|$. Find $\overrightarrow{AB}$ by subtracting coordinates, then use Pythagoras. This is the same as the distance formula from coordinate geometry.

A speed from a velocity vector. If an object moves by the same vector every second (or minute), the magnitude of that vector is the distance it covers each second: its speed.

An unknown component. If you are told the magnitude, write the Pythagoras equation and solve it. For $\left|\left(\begin{array}{c}k\\ 6\end{array}\right)\right|=10$: $k^2+36=100$, so $k^2=64$ and $k=8$ or $k=-8$. Both signs give a vector of length 10, so give both unless the question restricts $k$.