Number Concepts Study Guide

Learner Guide

Vectors: A Clear Guide for Beginners

Learn what vectors are, how they differ from scalars, how to add and subtract them, how to find magnitudes, and how to use vectors in geometry. This guide is designed to feel like a patient teacher walking you through the topic step by step.

Start Learning

How to use this guide

Read each section in order. Each idea is followed by examples and a short explanation of why the method works. If you are new to vectors, do not rush. Try the quick checks before moving on.

Big idea: A vector tells you how much and which direction. A scalar tells you only how much.

1. Vectors and scalars

A vector is a quantity with both magnitude and direction. A scalar has magnitude only.

Examples of scalars

  • Mass
  • Length
  • Area
  • Temperature
  • Speed
  • Time

Examples of vectors

  • Displacement
  • Velocity
  • Force
  • Acceleration
  • Translation

If a triangle is translated 4 units to the right and 2 units up, we can describe that movement by a vector. The vector contains both pieces of information: the size of the movement and its direction.

2. Column vectors

In two dimensions, a vector is often written as a column vector:

\[ \mathbf{a} = \begin{pmatrix} x \\ y \end{pmatrix} \]

The top entry shows horizontal movement. The bottom entry shows vertical movement.

Interpretation

  • x > 0: move right
  • x < 0: move left
  • y > 0: move up
  • y < 0: move down

Example

\[ \begin{pmatrix} 4 \\ 2 \end{pmatrix} \]

means move 4 units right and 2 units up.

Displacement from one point to another

\[\text{If point A moves to point B, then the vector from A to B is written as } \overrightarrow{AB} \text{. If A = (1, 3) and B = (5, 1), then:}\]

\[ \overrightarrow{AB} = \begin{pmatrix} 5 - 1 \\ 1 - 3 \end{pmatrix} = \begin{pmatrix} 4 \\ -2 \end{pmatrix} \]

\[\text{ This means that to get from A to B, move 4 units right and 2 units down.}\]

3. Magnitude of a vector

The length of a vector is called its magnitude or modulus.

\[ \left| \begin{pmatrix} x \\ y \end{pmatrix} \right| = \sqrt{x^2 + y^2} \]

This comes from Pythagoras' Theorem. The horizontal and vertical parts form the legs of a right-angled triangle.

Worked example

\[\text{ Find the magnitude of} \begin{pmatrix} -3 \\ 2 \end{pmatrix}\].

Step 1: Square each component.

\[ (-3)^2 = 9,\qquad 2^2 = 4 \]

Step 2: Add them.

\[ 9 + 4 = 13 \]

Step 3: Take the square root.

\[ \left| \begin{pmatrix} -3 \\ 2 \end{pmatrix} \right| = \sqrt{13} \]

4. Vector operations

4.1 Adding vectors

Add corresponding components.

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

Example:

\[ \begin{pmatrix} 4 \\ 2 \end{pmatrix} + \begin{pmatrix} 3 \\ -4 \end{pmatrix} = \begin{pmatrix} 7 \\ -2 \end{pmatrix} \]

4.2 Subtracting vectors

Again, subtract corresponding components.

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

4.3 Multiplying by a scalar

Multiply every component by the scalar.

\[ k\begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ka \\ kb \end{pmatrix} \]

Example:

\[ 4\begin{pmatrix} 3 \\ -2 \end{pmatrix} = \begin{pmatrix} 12 \\ -8 \end{pmatrix} \]

4.4 Reverse vectors

Reversing the direction of a vector changes its sign.

\[ \overrightarrow{BA} = -\overrightarrow{AB} \]

4.5 Linking points

If you move from A to B, then B to C, the combined movement is from A to C.

\[ \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} \]

5. Position vectors

\[\text{ A vector drawn from the origin O to a point is called a } \textbf{ position vector }\].

\[ \overrightarrow{OA} = \begin{pmatrix} x \\ y \end{pmatrix} \]

\[\text{ This means the point A has coordinates (x, y).}\]

Worked example

Let A = (3, 1) and B = (-1, -2).

\[ \overrightarrow{OA} = \begin{pmatrix} 3 \\ 1 \end{pmatrix}, \qquad \overrightarrow{OB} = \begin{pmatrix} -1 \\ -2 \end{pmatrix} \]

To find \[\overrightarrow{AB}\] subtract the position vectors:

\[ \overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = \begin{pmatrix} -1 \\ -2 \end{pmatrix} - \begin{pmatrix} 3 \\ 1 \end{pmatrix} = \begin{pmatrix} -4 \\ -3 \end{pmatrix} \]

Then the magnitude is:

\[ |\overrightarrow{AB}| = \sqrt{(-4)^2 + (-3)^2} = \sqrt{25} = 5 \]

6. Vectors in geometry

Vectors are powerful in geometry because they help us describe parallel lines, midpoints, diagonals, and shapes such as triangles and parallelograms.

6.1 Parallelograms

In a parallelogram, opposite sides are equal and parallel. If \[\overrightarrow{AB} = \mathbf{b} \text{ and } \overrightarrow{AD} = \mathbf{d}\] then:

\[ \overrightarrow{BC} = \mathbf{d},\qquad \overrightarrow{DC} = \mathbf{b} \]

So the diagonal AC can be written as:

\[ \overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC} = \mathbf{b} + \mathbf{d} \]

6.2 Midpoints

If M is the midpoint of a segment, then the vector to M is half of the full vector.

\[ \overrightarrow{AM} = \frac{1}{2}\overrightarrow{AB} \]

6.3 Collinearity

Points are collinear if one vector is a scalar multiple of another. For example, if

\[ \overrightarrow{AP} = 2\overrightarrow{AQ} \]

then \[\overrightarrow{AP} \text{ and } \overrightarrow{AQ}\] are in the same direction, so A , P, and Q lie on the same straight line.

6.4 A beginner-friendly geometry example

In a parallelogram ABCS, let \[\overrightarrow{AB}=\mathbf{b} \text{ and } \overrightarrow{AD}=\mathbf{d} \text{. If X is the midpoint of BC, find } \overrightarrow{AX} \]

Step 1: Since opposite sides of a parallelogram are equal,

\[ \overrightarrow{BC} = \overrightarrow{AD} = \mathbf{d} \]

Step 2: Because \(X\) is the midpoint of \(BC\),

\[ \overrightarrow{BX} = \frac{1}{2}\mathbf{d} \]

Step 3: Now add the route from \(A\) to \(B\) and from \(B\) to \(X\):

\[ \overrightarrow{AX} = \overrightarrow{AB} + \overrightarrow{BX} = \mathbf{b} + \frac{1}{2}\mathbf{d} \]

7. Common mistakes to avoid

Mistake

Confusing speed and velocity.

Fix

Speed is scalar. Velocity includes direction, so it is a vector.

Mistake

Subtracting coordinates in the wrong order.

Fix

For \[\overrightarrow{AB} \text{, always = } (B - A), not (A - B)\].

Mistake

Forgetting that reversing a vector changes the sign.

Fix

\[\overrightarrow{BA} = -\overrightarrow{AB}\].

Mistake

Adding magnitudes instead of components.

Fix

Add vectors component by component.

8. Try these

Try these questions. Click the button to reveal the answers.

  1. Is temperature a vector or a scalar ?
  2. What is \[\begin{pmatrix} 2 \\ -3 \end{pmatrix} + \begin{pmatrix} 5 \\ 4 \end{pmatrix} = ?\]
  3. Find the magnitude of \[\begin{pmatrix} 6 \\ 8 \end{pmatrix}\].
  4. If \[\overrightarrow{OA}=\begin{pmatrix}1\\2\end{pmatrix} \text{and } \overrightarrow{OB}=\begin{pmatrix}4\\7\end{pmatrix}\], find \[\overrightarrow{AB}\].

9. Practice tasks

Basic practice

  1. Classify each quantity as vector or scalar: time, force, speed, displacement.
  2. Find \[\overrightarrow{AB} \text{ if A=(2,1) and B=(7,-3)}\].
  3. Find the magnitude of \[\begin{pmatrix} -5 \\ 12 \end{pmatrix}\].
  4. Given \[\mathbf{a}=\begin{pmatrix}4\\7\end{pmatrix} \text{ and } \mathbf{b}=\begin{pmatrix}-3\\5\end{pmatrix} \text{,find }\mathbf{a}+\mathbf{b} \text{, } \mathbf{a}-\mathbf{b} \text{, and } 2\mathbf{a}\].

Geometry practice

  1. In parallelogram ABCD, \[\text{ if } \overrightarrow{AB}=\mathbf{b} \text{ and } \overrightarrow{AD}=\mathbf{d} \text{ , write } \overrightarrow{AC}\].
  2. If M is the midpoint of AB, \[\text{ write } \overrightarrow{AM} \text{ in terms of } \overrightarrow{AB}\].
  3. If \[\overrightarrow{PQ}= \begin{pmatrix}2\\6\end{pmatrix} \text{ and } \overrightarrow{PR}= \begin{pmatrix}1\\3\end{pmatrix} \text{ , explain why P, Q, and R are collinear.}\]

10. Formula sheet

\[ \begin{pmatrix} a \\ b \end{pmatrix} + \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a+c \\ b+d \end{pmatrix} \]
\[ \begin{pmatrix} a \\ b \end{pmatrix} - \begin{pmatrix} c \\ d \end{pmatrix} = \begin{pmatrix} a-c \\ b-d \end{pmatrix} \]
\[ k\begin{pmatrix} a \\ b \end{pmatrix} = \begin{pmatrix} ka \\ kb \end{pmatrix} \]
\[ \left|\begin{pmatrix} x \\ y \end{pmatrix}\right| = \sqrt{x^2+y^2} \]
\[ \overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} \]
\[ \overrightarrow{AB} + \overrightarrow{BC} = \overrightarrow{AC} \]

Vectors Interactive Quiz

Answer the questions. Correct answers trigger a Bravo modal. At the end, review worked solutions.

Question 1: Vectors or Scalars

Classify each quantity as a vector or a scalar.

Question 2: Position Vectors

Given:

  • \[\vec{OA}=\begin{pmatrix}3\\2\end{pmatrix}\]
  • \[\vec{OB}=\begin{pmatrix}5\\-2\end{pmatrix}\]
  • \[\vec{AC}=\begin{pmatrix}-3\\2\end{pmatrix}\]
  • \[\vec{BD}=\begin{pmatrix}-2\\-2\end{pmatrix}\]
  • \[\vec{CE}=\begin{pmatrix}-4\\-1\end{pmatrix}\]
  • \[\vec{DF}=\begin{pmatrix}1\\4\end{pmatrix}\]

Find \[\overrightarrow{EF}\] as a column vector.

Question 3: Vector Operations

Let \[\vec{a}=\begin{pmatrix}4\\7\end{pmatrix}\], \[\vec{b}=\begin{pmatrix}3\\-5\end{pmatrix}\], \[\vec{c}=\begin{pmatrix}0\\4\end{pmatrix}\]

Question 4: Parallelogram ABCD

In the parallelogram \[\textbf{ABCD , } \vec{AD}=\mathbf{d} \text{ and } \vec{AB}=\mathbf{b}\]. The point X is the midpoint of AB. The lines AC and DX intersect at Q.

Find \[\vec{AQ}\] in terms of \[\mathbf{a} \textbf{ and }\mathbf{b}\].

Question 5: Parallelogram OABC

\(OABC\) is a parallelogram.

\[ \vec{OA}=3\mathbf{p}-2\mathbf{q} \] \[ \vec{OC}=5\mathbf{p}+6\mathbf{q} \]

(a) Find \[\vec{AC}\].

Express your answer as simply as possible in terms of \[\mathbf{p} \text{ and } \mathbf{q}\].

(b) D is the point where \[\vec{BD}=-2\mathbf{p}+6\mathbf{q}\]. Using vector methods, show that D lies on the line AC produced.

Question 6: KLMN is a parallelogram

KLMN is a parallelogram with position vectors

\[ \vec{OK}=\begin{pmatrix}2\\-3\end{pmatrix},\quad \vec{OL}=\begin{pmatrix}6\\-3\end{pmatrix},\quad \vec{ON}=\begin{pmatrix}3\\5\end{pmatrix} \]

The point H lies on KM such that KH = HM

Question 7: ABCD is a quadrilateral

ABCD is a quadrilateral, not drawn to scale, with \[\vec{AB}=\mathbf{a} \text{ ,} \vec{BC}=\mathbf{b} \text{ and } \vec{AD}=2\vec{BC}\].

The point X divides BD in the ratio 3:2.