Geometry and measures - Vectors: apply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors (Foundation tier)

The idea

A vector has size and direction, so you can draw it as an arrow or write it as a column with the across movement on top and the up movement below. To add two vectors, add the tops and add the bottoms. On a diagram that is the arrow you get by following the first and then the second nose to tail, drawn from the start of the first to the end of the second. To subtract, add the reverse, and the reverse of a vector has the sign of both components changed. Multiplying by a positive scalar changes the length of the arrow without turning it, so doubling a vector doubles both components and keeps the direction, while a negative scalar reverses it. If a has top 3 and bottom −1, and b has top −2 and bottom 4, then a + b has top 1 and bottom 3, and 2a has top 6 and bottom −2.

What earns the marks

• A correct method, such as adding corresponding components, or drawing the second arrow starting from the head of the first.
• Each correct component of the answer, so a slip in one component still leaves the other creditable.
• A scalar multiple with both components multiplied, and the direction reversed when the scalar is negative.
• A resultant drawn on a grid with the correct start point, length and direction.

Common mistakes and how to put them right

• Subtracting the top components but copying the bottom component across unchanged.

• Changing the sign of only one component when reversing a vector.

• Multiplying only the top number by the scalar and leaving the bottom number as it was.

• Drawing the second arrow from the start of the first instead of from its head, which gives a resultant that goes nowhere near the right point.

Exam tips

• Set the two column vectors out one above the other in your working, tops lined up with tops. Cramped working is where component slips come from.
• A negative scalar turns the arrow through 180°, so check the direction of your answer against the diagram before you commit to it.
• When a diagram is given, label each arrow with its vector as you go. The marks come from the route you take, not from the picture itself.