Vector Magnitude (Length)

Statement

The magnitude, or length, of a vector tells us how long it is regardless of its direction. For a vector \((x,y)\), its magnitude is given by:

\[ \left\| \begin{pmatrix}x \ y\end{pmatrix} \right\| = \sqrt{x^2 + y^2} \]

This is the distance from the origin to the point \((x,y)\) in the coordinate plane.

Why it’s true

• A vector \((x,y)\) forms a right-angled triangle with sides of length \(|x|\) and \(|y|\).
• By Pythagoras’ theorem, the hypotenuse length is \(\sqrt{x^2+y^2}\).
• This works in all quadrants, because squaring removes negative signs.

Recipe (how to use it)

1. Write the components of the vector \((x,y)\).
2. Square each component.
3. Add the squares together.
4. Take the square root to find the magnitude.

Spotting it

Look for questions asking “find the length of a vector” or “calculate the magnitude of \((x,y)\)”.

Common pairings

• Often used before finding a unit vector.
• Linked with distance formula between two points.

Mini examples

1. Given: \((3,4)\). Find: magnitude. Answer: 5.
2. Given: \((5,12)\). Find: magnitude. Answer: 13.

Pitfalls

• Forgetting to square before adding.
• Forgetting to take the square root at the end.
• Using absolute values incorrectly — negatives are handled automatically by squaring.

Exam strategy

• Always double-check arithmetic inside the square root.
• Simplify square roots if possible (e.g. \(\sqrt{50}=5\sqrt{2}\)).
• Use exact surd form unless a decimal approximation is requested.

Summary

The magnitude of a vector is just the length of the line it represents, found by applying Pythagoras’ theorem to its components. This is one of the most commonly used vector results in GCSE mathematics and underpins many later topics.