Vectors

This lesson covers Vectors as required by the A-Level Mathematics Pure 1 specification. Vectors are quantities that have both magnitude (size) and direction, distinguishing them from scalars which have magnitude only. Vectors are used extensively in both pure mathematics and mechanics.

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Scalars and Vectors

Type	Definition	Examples
Scalar	A quantity with magnitude only	Speed, mass, temperature, distance
Vector	A quantity with magnitude and direction	Velocity, force, displacement, acceleration

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Vector Notation

Vectors can be written in several ways:

• As a column vector: components arranged vertically.
• As i-j notation: ai + bj where i is the unit vector in the x-direction and j is the unit vector in the y-direction.
• As a position vector: the vector from the origin O to a point A, written as OA.
• A bold letter: a, or an underlined letter: a̲.

Example: The vector from (0, 0) to (3, 4) can be written as 3i + 4j.

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Vector Operations

Addition and Subtraction

Vectors are added (or subtracted) component-wise: