Vector Addition, Subtraction, and Scalar Multiplication

Grade 6

Adding vectors

To add two vectors, add their corresponding components.

Geometrically, placing vectors tip to tail gives the resultant — the single vector that has the same effect as both combined.

On a diagram, the vector going from start to finish of the tip-to-tail chain is the resultant. If you travel along a then along b, the resultant is a + b.

Scalar multiplication

Multiplying a vector by a scalar scales it. 2a is twice as long as a, in the same direction. −3a is three times as long, in the opposite direction. All components get multiplied by the scalar.

Worked example

EXAMPLE

Vector a = (3, 1) and vector b = (−1, 4). Find 2a − b.

2a = (6, 2). 2a − b = (6 − (−1), 2 − 4) = (7, −2). Answer: (7, −2).

Vector paths

In geometry problems you often need to express one vector in terms of others. The rule: travel along known vectors, reversing direction (and sign) when you go against the arrow. This is the key technique for all vector proof questions.

Try this one

Past paper - 2023

OABC is a parallelogram.

\(\overrightarrow{OA}=a\) and \(\overrightarrow{OC}=c\)

The point N lies on line AB such that AN: NB = 3: 5.

(a) Find the following vectors in terms of a and c. Give your answers in their simplest form.

(i) \(\overrightarrow{OB}\)=------------[1]

(ii) \(\overrightarrow{ON}=\)----------- [2]

(b) Line CN is extended to reach point P, such that \(\overrightarrow{CP}=\frac85\overrightarrow{CN}\)

Show, using vectors, that OAP is a straight line. [4]

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