Vectors
Year 10 (IGCSE) 📐 Geometry & Measures Vector notation, addition, subtraction, scalar multiplication.
➡️ Vector Notation
A vector has both magnitude (size) and direction. It is written as a column vector $\begin{pmatrix}x\\y\end{pmatrix}$ or a bold letter such as $\mathbf{a}$.
Example: $\overrightarrow{AB} = \begin{pmatrix}3\\-2\end{pmatrix}$ means 3 units right and 2 units down from $A$ to $B$.
⚡ Magnitude
$$|\mathbf{v}| = \sqrt{x^2 + y^2}$$➕ Vector Operations
Add vectors tip-to-tail. Multiplying by a scalar changes the magnitude (and direction if negative).
⚡ Addition and Scalar Multiplication
$$\begin{pmatrix}a\\b\end{pmatrix} + \begin{pmatrix}c\\d\end{pmatrix} = \begin{pmatrix}a+c\\b+d\end{pmatrix} \qquad k\begin{pmatrix}a\\b\end{pmatrix} = \begin{pmatrix}ka\\kb\end{pmatrix}$$ $-\mathbf{a}$ has the same magnitude as $\mathbf{a}$ but points in the opposite direction.
🗺️ Vector Geometry
Express paths between points using given vectors, then use them to prove geometric properties.
Example: In parallelogram OABC, $\overrightarrow{OA} = \mathbf{a}$ and $\overrightarrow{OC} = \mathbf{c}$. Then $\overrightarrow{OB} = \mathbf{a} + \mathbf{c}$ and $\overrightarrow{AB} = \mathbf{c}$.
Vectors are parallel if one is a scalar multiple of the other. Use this to prove lines are parallel or that three points are collinear.
Ready to test yourself? Click the Quiz tab above to answer questions on this topic!
Interactive Demonstration — Vectors
➡️ Vector Calculator
Enter two vectors a and b, and scalar k.