Using a point O as a reference, the vector →OP that represents each point P is called the position vector of point P. Once the reference point O is fixed, points and position vectors correspond one-to-one.
Position Vectors
Reference point O, position vector of point P: →p = →OP.
If the position vectors of points A and B are →a and →b, respectively,
→AB = →b − →a (endpoint position − starting-point position)
Example ①
If the position vector of point A is →a = (2, 3) and that of B is →b = (5, −1), find the components of →AB.
Solution: →AB = →b − →a = (5−2, −1−3) = (3, −4)
If the position vector of point A is →a = (2, 3) and that of B is →b = (5, −1), find the components of →AB.
Solution: →AB = →b − →a = (5−2, −1−3) = (3, −4)
Distance Between Two Points
If the position vectors of points A and B are →a and →b,
AB = |→b − →a|
Example ②
Find AB when A(1, 2) and B(4, 6).
Solution: →AB=(3, 4)、|→AB|=√(9+16)=5
Find AB when A(1, 2) and B(4, 6).
Solution: →AB=(3, 4)、|→AB|=√(9+16)=5
Key Points
- Position vector: the vector from reference point O to each point
- →AB = →b − →a (endpoint − starting point)
- AB = |→b − →a| (distance)
Practice Problems
- When A(−1, 4) and B(3, 1), find the components and magnitude of →AB.
- For the three points O(0,0), A(2, 3), and B(5, 7), find the components of →OA, →OB, and →AB.
- When the position vectors of A and B are →a and →b, respectively, what does →a − →b represent?
Answers and Explanations
- Answer: →AB=(4, −3), magnitude 5
Explanation: (3−(−1), 1−4)=(4, −3)、√(16+9)=5. - Answer: →OA=(2,3)、→OB=(5,7)、→AB=(3,4)
Explanation: →AB=→OB−→OA. - Answer: →BA
Explanation: →a−→b=→OA−→OB=→BA. It is the vector from B to A.