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CAMBRIDGE O LEVEL · 2025–2027

Magnitude of a vector.

Use the components to calculate a vector’s length, while keeping direction separate.

Syllabus 7.3Visual notesWorked examples
BY THE END

Calculate magnitude · use modulus notation · connect scalar multiplication to length

01

Use Pythagoras on the components

A vector’s components describe two perpendicular changes. Its magnitude is the straight-line length of the combined displacement, so those changes can form the legs of a right-angled triangle.

Apply Pythagoras to that triangle: square the components, add them and take the non-negative square root. A negative component means movement in the opposite axis direction; it does not create a negative contribution to length after squaring.

Magnitude is the length of a vector. The horizontal and vertical components are perpendicular, so Pythagoras applies.

a = xy   →   |a| = √(x² + y²)
MAGNITUDE LAB · CHANGE THE COMPONENTS
-7-7-6-6-5-5-4-4-3-3-2-2-1-1112233445566770xyaO(3, 4)
Vector components 3 and 4. The horizontal and vertical changes form a right triangle; the vector length is 5.000.
|a| = √[(3)² + (4)²] = √25 = 5

Magnitude is non-negative. A displayed root can be simplified further if it contains a square factor.

WORKED EXAMPLE

Find the magnitude of −6−8.

√[(−6)² + (−8)²] = √100 = 10

Negative components indicate direction; their squares still contribute positively to length.

Pause and explainWhy can a vector with two negative components still have positive magnitude?

The component squares are non-negative. Magnitude measures length, not which direction the arrow points.

02

Modulus signs mean magnitude, not components

A vector and its magnitude are different kinds of mathematical object. The vector has direction and components; its magnitude is one non-negative number. Modulus bars tell you to take that length.

When a vector is scaled, its length scales by the absolute value of the scalar. Reversing an arrow changes its direction but not its length. Also, the length of a combined route is generally not the sum of the separate vector lengths, because directions affect the resultant.

|AB| or AB can denote the length of the segment from A to B, while AB denotes the directed vector. Follow the notation used in the question.

WORKED EXAMPLE

Find the magnitude of vector AB for A = (−1, 2), B = (3, 5).

AB = 43, so |AB| = √(4² + 3²) = 5
|ka| = |k| |a|

For example, if |a| = 5, then |−2a| = 10. A negative scalar reverses direction but does not make the length negative.

Pause and explainDo a and −a have different magnitudes?

No. Their directions are opposite but their arrow lengths are equal.

03

Your turn. Square before adding.

First find the vector components if the question supplies endpoints instead of a vector. Then square, add and root in separate steps.

Keep an exact root when requested and simplify any square factors. Check that magnitude is non-negative and that a scalar multiplication changes length by the absolute scalar.

1

Find the magnitude of 512.

Reveal worked answer
√(25 + 144) = 13.
2

Find the exact magnitude of 2−4.

Reveal worked answer
√(4 + 16) = √20 = 2√5.
3

A vector has magnitude 7. Find the magnitude of −3 times that vector.

Reveal worked answer
21. Use the absolute scalar: |−3| × 7 = 21.
4

Find the vector from A (2, 3) to B (−4, −5), then its magnitude.

Reveal worked answer
AB = −6−8, with magnitude 10.