Vectors in two and three dimensions, magnitude and direction, addition and scalar multiplication, position vectors, unit vectors, and geometric applications.

What this dot point is asking

Edexcel wants you to use vectors in two and three dimensions, write them in component or $\mathbf{i}$, $\mathbf{j}$, $\mathbf{k}$ form, find magnitudes and directions, add and subtract vectors and multiply by a scalar, use position vectors to find displacements, find unit vectors, and apply vectors to geometric problems such as proving points are collinear or finding a midpoint.

The answer

Components and magnitude

Adding, subtracting and scaling

Vectors add and subtract component by component: $(x_1\mathbf{i} + y_1\mathbf{j}) + (x_2\mathbf{i} + y_2\mathbf{j}) = (x_1 + x_2)\mathbf{i} + (y_1 + y_2)\mathbf{j}$. Multiplying a vector by a scalar $k$ multiplies each component by $k$, which stretches the vector by a factor $k$ and reverses it when $k$ is negative. Geometrically, addition is the triangle rule: place the tail of the second vector at the head of the first, and the resultant runs from the start to the finish.

Position vectors and displacement

The position vector of a point $A$ is $\overrightarrow{OA} = \mathbf{a}$ from the origin. The displacement from $A$ to $B$ is $\overrightarrow{AB} = \mathbf{b} - \mathbf{a}$, and the midpoint of $AB$ has position vector $\tfrac{1}{2}(\mathbf{a} + \mathbf{b})$.

Geometric applications

Examples in context

Try this

Q1. Find the magnitude of $\mathbf{a} = 2\mathbf{i} - 3\mathbf{j} + 6\mathbf{k}$. [2 marks]

• Cue. $|\mathbf{a}| = \sqrt{4 + 9 + 36} = \sqrt{49} = 7$.

Q2. Points $A$ and $B$ have position vectors $\mathbf{i} + 2\mathbf{j}$ and $5\mathbf{i} - 2\mathbf{j}$. Find $\overrightarrow{AB}$ and its magnitude. [3 marks]

• Cue. $\overrightarrow{AB} = 4\mathbf{i} - 4\mathbf{j}$, magnitude $\sqrt{32} = 4\sqrt{2}$.