Vectors and magnitude

• A vector has size and direction. It is written as a column vector xy, as ⟶AB, or as a bold letter a.
• Add and subtract vectors by working on the top numbers and the bottom numbers separately.
• Multiplying by a scalar k stretches the vector by k; a negative k also reverses it.
• The magnitude of xy is √(x² + y²). It is written |a| or |⟶AB|. This formula is not on the List of formulas.

Three ways to write the same thing

A vector has a magnitude (size) and a direction. A scalar has size only, like 5 or −2. The exam prints vectors in three ways:

⟶BA is the same journey as ⟶AB in reverse, so ⟶BA = −⟶AB. Two vectors are equal when they have the same size and direction, wherever they are drawn.

One movement after another

Adding vectors means doing one movement and then the next: ⟶AB + ⟶BC = ⟶AC. On paper, draw them “nose to tail”; the sum (the resultant) goes from the start of the first to the end of the last. Subtracting a vector is adding its reverse: a − b = a + (−b).

Stretch, shrink, reverse

To multiply a vector by a number, multiply both components. ka is parallel to a and k times as long. If k is negative, ka points the opposite way.

Pythagoras on the two components

The magnitude of a vector is its length. The two components are the shorter sides of a right-angled triangle, so Pythagoras gives the length. The exam writes it with modulus signs: |a| is the magnitude of a, and |⟶AB| is the magnitude of ⟶AB (the distance AB).