VectorsEdexcel 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.

Column vector notation is the standard form used in GCSE maths:

• A vector can be written as a column with two components: $\begin{pmatrix} x \\ y \end{pmatrix}$
• The top number represents horizontal movement (right is positive, left is negative)
• The bottom number represents vertical movement (up is positive, down is negative)

Alternative notations you may encounter:

• Bold letters: a or b
• Underlined letters: a or b
• Arrow notation: $\overrightarrow{AB}$ (vector from point A to point B)

Key points:

• The vector $\begin{pmatrix} 3 \\ 2 \end{pmatrix}$ means move 3 units right and 2 units up
• The vector $\begin{pmatrix} -2 \\ 5 \end{pmatrix}$ means move 2 units left and 5 units up
• Equal vectors have the same magnitude and direction, regardless of starting position

Adding vectors means combining their effects. When you add vectors, you perform the operation component-by-component:

$\begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a+c \\ b+d \end{pmatrix}$

Subtracting vectors works the same way:

$\begin{pmatrix} a \\ b \end{pmatrix} - \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a-c \\ b-d \end{pmatrix}$

Step-by-step process for vector addition:

1. Add the x-components (top numbers) together
2. Add the y-components (bottom numbers) together
3. Write the result as a new column vector

Step-by-step process for vector subtraction:

1. Subtract the x-components
2. Subtract the y-components
3. Write the result as a new column vector

Important property: Vector addition is commutative (order doesn't matter): $\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}$

Scalar multiplication means multiplying a vector by an ordinary number (a scalar). This changes the magnitude of the vector but keeps the direction the same (unless the scalar is negative).

Formula for scalar multiplication: $k \begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ka \\ kb \end{pmatrix}$

where $k$ is any scalar (positive or negative number).

What happens:

• If $k > 1$: the vector becomes longer
• If $0 < k < 1$: the vector becomes shorter
• If $k = 0$: the vector becomes the zero vector $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$
• If $k < 0$: the vector reverses direction and changes magnitude

Step-by-step process:

1. Multiply the scalar by the x-component
2. Multiply the scalar by the y-component
3. Write the result as a new column vector

Practical use: Parallel vectors can be expressed as scalar multiples of each other. If $\mathbf{b} = k\mathbf{a}$ for some scalar $k$, then the vectors are parallel.

The magnitude of a vector is its length. For a vector $\mathbf{v} = \begin{pmatrix} a \\ b \end{pmatrix}$, we use the Pythagoras theorem to find the magnitude.

Formula for magnitude: $|\mathbf{v}| = |\begin{pmatrix} a \\ b \end{pmatrix}| = \sqrt{a^2 + b^2}$

The notation $|\mathbf{v}|$ or $\|\mathbf{v}\|$ both mean 'the magnitude of vector v'.

Step-by-step process:

1. Square the x-component
2. Square the y-component
3. Add the squares together
4. Take the square root of the sum

Key point: The magnitude is always positive (or zero for the zero vector), even if the vector components are negative.

Example calculations:

• The magnitude of $\begin{pmatrix} 3 \\ 4 \end{pmatrix}$ is $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$
• The magnitude of $\begin{pmatrix} -5 \\ 12 \end{pmatrix}$ is $\sqrt{(-5)^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13$

A position vector describes the location of a point relative to a fixed origin. If the origin is at O and point A is at position $\begin{pmatrix} 3 \\ 5 \end{pmatrix}$, then the position vector of A is $\overrightarrow{OA} = \begin{pmatrix} 3 \\ 5 \end{pmatrix}$.

Finding the vector between two points: To find the vector from point A to point B, subtract the position vector of A from the position vector of B:

$\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$

Alternatively, if A has coordinates $(x_1, y_1)$ and B has coordinates $(x_2, y_2)$:

$\overrightarrow{AB} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix}$

Describing paths using vectors:

• A path from O to A to B can be written as: $\overrightarrow{OA} + \overrightarrow{AB} = \overrightarrow{OB}$
• This is the triangle law of vector addition: the sum of two sides of a triangle equals the third side
• You can break any journey into vector segments and add them together

Application: To find the position of a point C such that $\overrightarrow{AC} = k\overrightarrow{AB}$ (a point on the line from A to B), use: $\overrightarrow{OC} = \overrightarrow{OA} + k\overrightarrow{AB}$

Vectors are powerful tools for proving geometric properties without coordinates. Key techniques include:

Proving lines are parallel: Two lines are parallel if their direction vectors are scalar multiples of each other. If $\overrightarrow{AB} = k\overrightarrow{CD}$ for some scalar $k$, then AB is parallel to CD.

Proving points are collinear: Three points A, B, C are collinear (on the same line) if one vector can be expressed as a scalar multiple of another. For example, if $\overrightarrow{AB} = m\overrightarrow{BC}$, the points are collinear.

Finding the midpoint of a line segment: The position vector of the midpoint M of AB is: $\overrightarrow{OM} = \frac{1}{2}(\overrightarrow{OA} + \overrightarrow{OB})$

Proving equal lengths: Two line segments have equal length if their magnitude vectors are equal: $|\overrightarrow{AB}| = |\overrightarrow{CD}|$

General proof strategy:

1. Express all required vectors in terms of given vectors
2. Use vector algebra to manipulate expressions
3. Show the required relationship (e.g., one vector is a scalar multiple of another)
4. State the geometric conclusion clearly

Common proof structures:

• To prove ABCD is a parallelogram: show $\overrightarrow{AB} = \overrightarrow{DC}$ and $\overrightarrow{AD} = \overrightarrow{BC}$
• To prove a quadrilateral is a rhombus: show all four sides have equal magnitude
• To prove perpendicularity (Higher Tier concepts): vectors are perpendicular if their dot product equals zero