In vector geometry, routes are written in terms of given vectors. If OA = a and OB = b, then AB = AO + OB = −a + b = b − a. Always travel along known vectors to get from start to finish.
The midpoint M of AB is half way along: OM = OA + ½AB = a + ½(b − a) = ½(a + b). To show three points are on a straight line, show that two of the vectors between them are multiples of each other and share a point.
Worked example
OA = a and OB = b. M is the midpoint of AB. Find OM.
- AB = b − a, so AM = ½(b − a).
- OM = a + ½(b − a) = ½a + ½b.
Answer:½(a + b)
Key words
- route
- a path along known vectors
- midpoint
- half way along a segment
- collinear
- on the same straight line
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