Teach Me Maths

Vectors

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How to read and write column vectors, add and subtract them, multiply them by a number, draw them on a grid and write simple paths between points.

What a vector is

A vector describes a movement: how far, and in which direction. A column vector writes the movement as two numbers, one above the other. For example, \begin{pmatrix}-4\\1\end{pmatrix} means 4 to the left and 1 up.

  • The top number is the movement across: positive is right, negative is left.
  • The bottom number is the movement up or down: positive is up, negative is down.

There is no line between the two numbers, so a column vector is not a fraction. It is not a pair of coordinates either: coordinates give a position, but a vector gives a movement.

Vectors are named by letters, printed in bold in the exam, such as \mathbf{a} and \mathbf{b}. When you write them by hand, underline them: \underline{a}. The vector from the point A to the point B is written \overrightarrow{AB}.

In the exam you are usually asked to work out something like 2\mathbf{a}+\mathbf{b} as a column vector, for 2 marks, near the end of the paper. You may also be asked to write down a vector drawn on a grid, to draw a vector such as 2\mathbf{d} from a point, or to write a path such as \overrightarrow{AB} in terms of \mathbf{a} and \mathbf{b}. Using a vector to move a shape is on the Translations and Column Vectors page.

Vectors on a grid

Count from the start of the arrow to the arrowhead: across first, then up or down. Here \mathbf{a} goes 4 right and 2 up, so \mathbf{a}=\begin{pmatrix}4\\2\end{pmatrix}. \mathbf{b} goes 2 left and 3 up, so \mathbf{b}=\begin{pmatrix}-2\\3\end{pmatrix}

A vector can start anywhere: two arrows with the same length and the same direction are the same vector. When you draw a vector, always put the arrowhead on the end, because the arrowhead shows the direction.

Direction matters. Going the other way along a vector, from B back to A, gives an arrow the same length pointing the opposite way, so both signs change. If \overrightarrow{AB}=\begin{pmatrix}4\\-3\end{pmatrix}, then \overrightarrow{BA}=\begin{pmatrix}-4\\3\end{pmatrix}

Worked example 1: Writing a vector from a grid

The vector \mathbf{v} is drawn on the grid. Write \mathbf{v} as a column vector.

  1. Start at the dot, the start of the arrow, and count to the arrowhead. Count across first.
  2. The arrow goes 5 squares to the right, so the top number is 5

  3. Your turn. The arrow goes 3 squares down. What is the bottom number of the column vector?

    Down is negative, so the bottom number is -3

  4. Answer: \mathbf{v}=\begin{pmatrix}5\\-3\end{pmatrix}

Adding and subtracting vectors

To add two column vectors, add the top numbers, then add the bottom numbers:

\begin{pmatrix}2\\5\end{pmatrix}+\begin{pmatrix}4\\-3\end{pmatrix}=\begin{pmatrix}2+4\\5+(-3)\end{pmatrix}=\begin{pmatrix}6\\2\end{pmatrix}

To subtract, subtract the top numbers, then subtract the bottom numbers.

Keep the top numbers and the bottom numbers apart. Never add a top number to a bottom number, and never treat a column vector like a fraction. The answer is another column vector, with two numbers, not a single number.

On a grid, \mathbf{a}+\mathbf{b} means: follow \mathbf{a}, then follow \mathbf{b} from where \mathbf{a} ends. The arrow from the very start to the very end is \mathbf{a}+\mathbf{b}. With \mathbf{a}=\begin{pmatrix}4\\2\end{pmatrix} and \mathbf{b}=\begin{pmatrix}-2\\3\end{pmatrix} it goes 2 right and 5 up, the same as adding the numbers: 4+(-2)=2 and 2+3=5

Worked example 2: Subtracting with negative numbers

\mathbf{a}=\begin{pmatrix}6\\-1\end{pmatrix} and \mathbf{b}=\begin{pmatrix}-2\\4\end{pmatrix}. Work out \mathbf{a}-\mathbf{b} as a column vector.

  1. Subtract the top numbers and the bottom numbers separately: \mathbf{a}-\mathbf{b}=\begin{pmatrix}6-(-2)\\-1-4\end{pmatrix}

  2. Your turn. Work out the top number, 6-(-2).

    Subtracting -2 is the same as adding 2: 6-(-2)=6+2=8

  3. The bottom number is -1-4=-5
  4. Answer: \mathbf{a}-\mathbf{b}=\begin{pmatrix}8\\-5\end{pmatrix}

Multiplying a vector by a number

3\mathbf{a} means \mathbf{a}+\mathbf{a}+\mathbf{a}, so multiply both numbers by 3:

3\begin{pmatrix}4\\-1\end{pmatrix}=\begin{pmatrix}3\times4\\3\times(-1)\end{pmatrix}=\begin{pmatrix}12\\-3\end{pmatrix}

The number you multiply by is called a scalar. Multiplying by 2 gives an arrow in the same direction, twice as long. Multiplying by -1 turns the arrow round, so -\mathbf{a} is the same length as \mathbf{a} but points the opposite way.

Multiplying by a negative number changes both signs: -2\begin{pmatrix}3\\-5\end{pmatrix}=\begin{pmatrix}-6\\10\end{pmatrix}. Always multiply, never square: 2\mathbf{b} means double each number in \mathbf{b}.

Worked example 3: Multiplying, then subtracting

\mathbf{a}=\begin{pmatrix}2\\-3\end{pmatrix} and \mathbf{b}=\begin{pmatrix}5\\-1\end{pmatrix}. Work out 4\mathbf{a}-2\mathbf{b} as a column vector.

  1. Multiply first: 4\mathbf{a}=\begin{pmatrix}4\times2\\4\times(-3)\end{pmatrix}=\begin{pmatrix}8\\-12\end{pmatrix}

  2. Your turn. Work out 2\mathbf{b}. Enter it as (top, bottom).

    Write your answer as (x, y)

    2\mathbf{b}=\begin{pmatrix}10\\-2\end{pmatrix}

  3. Now subtract: 4\mathbf{a}-2\mathbf{b}=\begin{pmatrix}8-10\\-12-(-2)\end{pmatrix}
  4. Top: 8-10=-2. Bottom: -12-(-2)=-12+2=-10
  5. Answer: 4\mathbf{a}-2\mathbf{b}=\begin{pmatrix}-2\\-10\end{pmatrix}

Worked example 4: Drawing a vector from a point

\mathbf{p}=\begin{pmatrix}2\\3\end{pmatrix} and \mathbf{q}=\begin{pmatrix}4\\-1\end{pmatrix}. On a coordinate grid, draw the vector 2\mathbf{q}-\mathbf{p}, starting at the point P(-3,4).

  1. Work out the column vector first: 2\mathbf{q}=\begin{pmatrix}8\\-2\end{pmatrix}
  2. 2\mathbf{q}-\mathbf{p}=\begin{pmatrix}8-2\\-2-3\end{pmatrix}=\begin{pmatrix}6\\-5\end{pmatrix}
  3. So the arrow goes 6 squares right and 5 squares down from P.

  4. Your turn. P is at (-3,4). Where does the arrow end? Give the coordinates of the arrowhead.

    Write your answer as (x, y)

    The arrowhead is at (-3+6,\ 4-5)=(3,-1)

  5. Answer: draw a straight line from P to (3,-1), with the arrowhead at (3,-1)

Paths between points

Going from A to B, then from B to C, gets you from A to C:

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

Going backwards changes the sign: \overrightarrow{BA}=-\overrightarrow{AB}. So if \overrightarrow{AB}=\mathbf{a}, then \overrightarrow{BA}=-\mathbf{a}

In a parallelogram, opposite sides are parallel and the same length. So if \overrightarrow{AB}=\mathbf{a}, then \overrightarrow{DC}=\mathbf{a} too, because it points the same way. In the same way, \overrightarrow{BC}=\overrightarrow{AD}=\mathbf{b}.

To write a path in terms of \mathbf{a} and \mathbf{b}, go along sides you know, write down each vector with the right sign, then collect like terms. For example, \overrightarrow{AC}=\overrightarrow{AB}+\overrightarrow{BC}=\mathbf{a}+\mathbf{b}

Worked example 5: Paths through the middle of a parallelogram

PQRS is a parallelogram. Its diagonals cross at M. \overrightarrow{PM}=\mathbf{p} and \overrightarrow{MQ}=\mathbf{q}. Write \overrightarrow{QR} in terms of \mathbf{p} and \mathbf{q}.

  1. The diagonals of a parallelogram cross halfway along each one, so M is the midpoint of PR and of QS.
  2. \overrightarrow{MR} carries on along the same diagonal as \overrightarrow{PM}, in the same direction and the same length, so \overrightarrow{MR}=\mathbf{p}

  3. Your turn. \overrightarrow{QM} goes along \overrightarrow{MQ} backwards. Write \overrightarrow{QM} in terms of \mathbf{q}.

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    Going backwards changes the sign: \overrightarrow{QM}=-\mathbf{q}

  4. \overrightarrow{QR}=\overrightarrow{QM}+\overrightarrow{MR}=-\mathbf{q}+\mathbf{p}
  5. Answer: \overrightarrow{QR}=\mathbf{p}-\mathbf{q}

Worked example 6: A path around a trapezium

ABCD is a trapezium. \overrightarrow{AB}=\mathbf{a}, \overrightarrow{AD}=\mathbf{b} and \overrightarrow{DC}=2\mathbf{a}. Write \overrightarrow{BC} in terms of \mathbf{a} and \mathbf{b}. Simplify your answer.

  1. We don't know BC directly, so go the long way round: from B to A, then A to D, then D to C.
  2. \overrightarrow{BA}=-\mathbf{a}, because it goes along \overrightarrow{AB} backwards.
  3. \overrightarrow{BC}=\overrightarrow{BA}+\overrightarrow{AD}+\overrightarrow{DC}=-\mathbf{a}+\mathbf{b}+2\mathbf{a}

  4. Your turn. Simplify -\mathbf{a}+\mathbf{b}+2\mathbf{a}

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    -\mathbf{a}+2\mathbf{a}=\mathbf{a}, so \overrightarrow{BC}=\mathbf{a}+\mathbf{b}

  5. Answer: \overrightarrow{BC}=\mathbf{a}+\mathbf{b}

Common mistakes

  • Treating a column vector like a fraction. There is no dividing line: work on the top numbers and the bottom numbers separately, and never cancel.
  • Giving a single number as the answer. Adding, subtracting or multiplying column vectors always gives another column vector.
  • Slipping with negative numbers. Write each sum out, such as -1-4=-5 and 6-(-2)=8, before you write the answer.
  • Squaring instead of multiplying. 2\mathbf{b} means double both numbers in \mathbf{b}.
  • Swapping the numbers or going the wrong way. Count from the start of the arrow to the arrowhead, across first, with right and up positive.
  • Drawing a vector with no arrowhead. Without the arrowhead the direction is lost, and the mark with it.

Practice questions

More on this topic: Vectors worksheet with full solutions