Vectors: column vectors, adding, subtracting and multiplying vectors by a scalar, the magnitude of a vector, and using vectors in geometric proofs including parallel lines and points lying on a straight line (Higher tier).

What this dot point is asking

A vector has both size (magnitude) and direction. Edexcel expects you to write vectors as column vectors, add, subtract and scale them, find a vector's magnitude, and at Higher tier use vectors in geometric proofs, such as showing that two lines are parallel or that three points lie on a straight line. Vector proof is a discriminating Higher topic that rewards clear, route-based reasoning.

Column vectors and arithmetic

A column vector records a movement: the top number is the horizontal change, the bottom number the vertical change.

So $\begin{pmatrix} 5 \\ 2 \end{pmatrix} + \begin{pmatrix} -1 \\ 3 \end{pmatrix} = \begin{pmatrix} 4 \\ 5 \end{pmatrix}$, and $3\begin{pmatrix} 2 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ -3 \end{pmatrix}$. The negative of a vector reverses its direction: $-\mathbf{a}$ points the opposite way to $\mathbf{a}$.

Magnitude of a vector

The magnitude is the length of the vector, found with Pythagoras because the components form a right-angled triangle.

The magnitude of $\begin{pmatrix} x \\ y \end{pmatrix}$ is $\sqrt{x^2 + y^2}$. So the vector $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ has magnitude $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$. This links directly to the distance between two points.

Vectors in geometry

Vectors describe journeys between labelled points. The notation $\overrightarrow{AB}$ means the vector from $A$ to $B$, and a key rule is $\overrightarrow{AB} = -\overrightarrow{BA}$. To find a vector between two points, travel via the origin or any known route: $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = \mathbf{b} - \mathbf{a}$.

Proving parallel and collinear (Higher)

Two vectors are parallel if one is a scalar multiple of the other, that is $\overrightarrow{PQ} = k\,\overrightarrow{RS}$ for some number $k$; the direction is the same (or exactly opposite). Three points are collinear (lie on one straight line) if the vector between two of them is a scalar multiple of the vector between another pair and they share a common point. A proof states the vectors, shows one is a multiple of the other, and concludes with the named property, for example "so $\overrightarrow{AB}$ is parallel to $\overrightarrow{CD}$".

Setting out a vector proof

A clean vector proof reads like a short argument. First express the two vectors you want to compare in terms of the base vectors $\mathbf{a}$ and $\mathbf{b}$, simplifying each fully. Then factorise to reveal a common factor: if you can write $\overrightarrow{PQ} = k\,\overrightarrow{RS}$, the two are parallel. Finally write a concluding sentence naming what you have shown and, for collinearity, point out that the two parallel segments share a common point, so the three points must lie on one straight line. Because the marks are split between the vector work and the conclusion, never stop at the algebra; the examiner wants the geometric statement that the algebra proves.

Try this

Q1. Find the magnitude of the vector $\begin{pmatrix} 5 \\ 12 \end{pmatrix}$. [2 marks]

• Cue. $\sqrt{5^2 + 12^2} = \sqrt{169} = 13$.

Q2. $\mathbf{p} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}$ and $\mathbf{q} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$. Work out $\mathbf{p} + 2\mathbf{q}$ as a column vector. [2 marks]

• Cue. $2\mathbf{q} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}$, so $\mathbf{p} + 2\mathbf{q} = \begin{pmatrix} 7 \\ 0 \end{pmatrix}$.