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MathematicsGCSE Higher & A Level

What is vector?

A vector is a quantity with both magnitude and direction, such as displacement, velocity or force.

Vector explained

Vectors add tip-to-tail, and multiplying by a scalar changes length but not direction unless the scalar is negative. Parallel vectors are scalar multiples of each other.
In geometry proofs, express every route through the shape in terms of two base vectors, then show two routes are multiples of one another to prove points are collinear.

Key formula

∣a∣=x2+y2|\mathbf{a}| = \sqrt{x^{2} + y^{2}}

Worked example

If a=(34)\mathbf{a} = \binom{3}{4} then ∣a∣=5|\mathbf{a}| = 5 and 2a=(68)2\mathbf{a} = \binom{6}{8} points the same way but twice as far.

Examiner tip

Underline or embolden vector letters in your working. Examiners look for clear vector notation when awarding the communication mark.

Vector: common questions

What is the difference between a vector and a scalar?
A scalar has size only, such as mass or speed. A vector has size and direction, such as weight or velocity.
How do I prove two vectors are parallel?
Show one is a scalar multiple of the other; if they also share a point, the three points concerned are collinear.

Related terms

Gradient

The gradient of a line measures its steepness — the change in y divided by the change in x between two points.

Pythagoras' theorem

Pythagoras' theorem states that in a right-angled triangle the square of the hypotenuse equals the sum of the squares of the other two sides.

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