VectorsAQA GCSE Maths: Revision notes

A vector is a quantity that has both magnitude (size) and direction. Vectors are different from scalars, which only have magnitude.

Vectors can be represented in several ways:

• Column vector notation: A vector from point A to point B is written as a column with two numbers. The top number represents the horizontal displacement (movement right/left), and the bottom number represents the vertical displacement (movement up/down). For example, $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ means move 3 units right and 2 units down.

• Bold or arrow notation: Vectors are often written as a, $\vec{AB}$, or $\mathbf{a}$. The arrow from A to B indicates direction and magnitude.

• Equal vectors: Two vectors are equal if they have the same magnitude and direction, regardless of their starting position.

For example, the vector from (1, 2) to (4, 5) equals $\begin{pmatrix} 3 \\ 3 \end{pmatrix}$, as does the vector from (0, 0) to (3, 3).

Adding vectors: To add two vectors, add their corresponding components.

If a = $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and b = $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$, then a + b = $\begin{pmatrix} 2+1 \\ 3+(-1) \end{pmatrix}$ = $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$

Geometrically, place the vectors end-to-end in sequence; the resultant vector connects the start of the first to the end of the last.

Subtracting vectors: Subtract corresponding components, or add the negative vector.

a - b = $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ - $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$ = $\begin{pmatrix} 2-1 \\ 3-(-1) \end{pmatrix}$ = $\begin{pmatrix} 1 \\ 4 \end{pmatrix}$

Multiplying a vector by a scalar: Multiply each component of the vector by the scalar (a number).

If a = $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$, then 3a = $\begin{pmatrix} 6 \\ 9 \end{pmatrix}$

A negative scalar reverses the direction: -2a = $\begin{pmatrix} -4 \\ -6 \end{pmatrix}$

Scalar multiplication makes vectors parallel: if b = ka (where k is a non-zero number), then a and b are parallel.

The magnitude (or modulus) of a vector is its length. For a column vector $\begin{pmatrix} x \\ y \end{pmatrix}$, the magnitude is found using Pythagoras' theorem:

|a| = √(x² + y²)

For example, if a = $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$, then |a| = √(3² + 4²) = √(9 + 16) = √25 = 5

This works because the vector components form a right-angled triangle with the vector as the hypotenuse.

Finding unit vectors: A unit vector has magnitude 1. To find the unit vector in the direction of a, divide a by its magnitude:

Unit vector = a / |a|

For a = $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ (magnitude 5), the unit vector is $\begin{pmatrix} 3/5 \\ 4/5 \end{pmatrix}$ or $\begin{pmatrix} 0.6 \\ 0.8 \end{pmatrix}$

Vectors can be used to describe journeys or paths between points. If you know the vectors connecting intermediate points, you can combine them (using vector addition) to find the overall displacement.

Key notation: The vector from point A to point B is written as $\vec{AB}$ or AB.

Multi-step paths: To find the total displacement from A to C via B, add the vectors:

$\vec{AC}$ = $\vec{AB}$ + $\vec{BC}$

Example: If a person walks from A to B with displacement $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$, then from B to C with displacement $\begin{pmatrix} 1 \\ 4 \end{pmatrix}$, the total displacement from A to C is:

$\vec{AC}$ = $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ + $\begin{pmatrix} 1 \\ 4 \end{pmatrix}$ = $\begin{pmatrix} 4 \\ 6 \end{pmatrix}$

Reverse paths: To go backwards, negate the vector. If $\vec{AB}$ = $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$, then $\vec{BA}$ = $\begin{pmatrix} -3 \\ -2 \end{pmatrix}$

Closed loops: If a path returns to the starting point, the sum of all displacement vectors equals zero: $\vec{AB}$ + $\vec{BC}$ + $\vec{CA}$ = 0

Vectors are powerful tools for proving geometric properties without coordinates. Common applications include proving parallelograms, collinearity, and equal lengths.

Proving a quadrilateral is a parallelogram: A quadrilateral ABCD is a parallelogram if opposite sides are parallel and equal. Check if $\vec{AB}$ = $\vec{DC}$ and $\vec{AD}$ = $\vec{BC}$.

Proving points are collinear: Three points A, B, and C are collinear (on the same line) if $\vec{AB}$ = k$\vec{AC}$ for some scalar k. In other words, one vector is a scalar multiple of the other.

Example: If $\vec{AB}$ = $\begin{pmatrix} 2 \\ 3 \end{pmatrix}$ and $\vec{AC}$ = $\begin{pmatrix} 4 \\ 6 \end{pmatrix}$, then $\vec{AC}$ = 2$\vec{AB}$, so A, B, and C are collinear.

Expressing position vectors: If point M is the midpoint of AB, then $\vec{AM}$ = ½$\vec{AB}$. More generally, if a point divides a line segment in a ratio, vectors allow us to express its position precisely.

Key proof strategy:

1. Express all required vectors in terms of given vectors (e.g., a and b)
2. Simplify using vector algebra
3. Interpret the result geometrically (e.g., 'vectors are equal, so the lines are parallel')
4. State the conclusion clearly