Position vectors and constructing points
A position vector locates one point from the fixed origin O. A displacement is a free vector: moving its arrow without turning or resizing it does not change the vector. A position vector is fixed at O, and its components are the coordinates of the point.
Keep position and displacement distinct
Position starts at O; displacement starts at A
end position − start position
- starts at O and ends at A.
- compares two positions.
- The midpoint M of AB has position vector .
Worked example
Midpoint from two position vectors
If and , average matching coordinates.
Examiner note
Translate a shape statement into arrows
Worked example
Current-paper trapezium relation
Let , and . The condition is .
A trapezium has a pair of parallel opposite sides. Here the factor 3 says that DC points in the same direction as AB and is three times its magnitude.
Since :
(9709/32/O/N/24 Q9(a))
Common mistake
Build an unknown vertex from a vector relation
A, B, C and D are consecutive vertices of a parallelogram. Given , and , find D.
Show worked answer
“Consecutive” means listed in order around the shape. Opposite sides of a parallelogram are equal and parallel, so .