Use vectors in mechanics: add and subtract vectors, multiply by a scalar, use component (i and j) form, and find the magnitude and direction of a vector.

What this dot point is asking

A vector has both size and direction, and CCEA GCSE Further Mathematics uses vectors throughout the Mechanics unit to represent displacement, velocity, acceleration and force. You must add and subtract vectors, multiply them by scalars, work in component ($\mathbf{i}$ and $\mathbf{j}$) form, and find the magnitude and direction of a vector. These skills let you combine forces and describe motion in two dimensions cleanly.

Component form

Vectors are most easily handled in component form, splitting each into a part along $\mathbf{i}$ (horizontal) and a part along $\mathbf{j}$ (vertical). A displacement of $5\mathbf{i} + 3\mathbf{j}$ means $5$ units across and $3$ units up. Writing every vector this way turns vector arithmetic into ordinary arithmetic on the separate components.

Adding, subtracting and scaling

Vector addition and subtraction work component by component, keeping $\mathbf{i}$ parts with $\mathbf{i}$ parts and $\mathbf{j}$ parts with $\mathbf{j}$ parts. Scalar multiplication multiplies both components by the number.

So $(2\mathbf{i} + 3\mathbf{j}) + (4\mathbf{i} - \mathbf{j}) = 6\mathbf{i} + 2\mathbf{j}$, and $3(2\mathbf{i} - \mathbf{j}) = 6\mathbf{i} - 3\mathbf{j}$. Adding force vectors gives the resultant force, the single force equivalent to several acting together, which is central to the forces topic.

Magnitude and direction

The magnitude of a vector is its length, found by applying Pythagoras to its components. The direction is the angle the vector makes, usually measured from the positive horizontal axis.

Why this matters

Vectors are the language of two-dimensional mechanics. Resolving and combining forces relies on component form, projectile motion splits velocity into horizontal and vertical components, and momentum is a vector too. The magnitude formula reuses Pythagoras from coordinate geometry, and the direction calculation reuses the trigonometry from the Pure unit, so this topic ties together skills you already have and applies them to motion and force.