Worked example 1
Find a route vector
A has position vector (2, 1) and B has position vector (7, 4). Find AB.
- Use AB = b − a.
- Subtract corresponding components.
Answer: AB = (5, 3)
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Higher Mathematics
Position vectors and vector geometry
Use position vectors, routes and scalar multiples to prove geometric facts.
Before you start
- Use vector addition and subtraction component by component.
- Understand position vectors from the origin.
- Know that parallel vectors are scalar multiples.
Explanation
Position vectors locate points from the origin. Vector geometry uses routes between points to prove facts such as parallel lines, collinearity and ratios.
A clear route statement is often the difference between a convincing proof and a list of components.
Visual support
Method and rules
- If position vectors are a and b, then AB = b − a.
- Parallel vectors are scalar multiples.
- Points are collinear when connecting vectors are parallel and share a point.
Worked examples
Worked example 2
Show parallel vectors
u = (2, −1) and v = (6, −3)
- Compare components.
- v is 3 times u.
So: The vectors are parallel because v = 3u.
Watch out
- Using AB = a − b instead of b − a
- Confusing a point's position vector with a route vector.
- Claiming vectors are equal when they are only parallel.
- Not stating the geometric conclusion after the calculation.
Exam reminder
For proof questions, write the vector relationship and the conclusion in words, such as 'therefore AB is parallel to CD'.