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Relative to a fixed origin, points P, Q and R have position vectors p, q and r respectively - Edexcel - A-Level Maths Pure - Question 4 - 2020 - Paper 2
Question 4
Relative to a fixed origin, points P, Q and R have position vectors p, q and r respectively. Given that - P, Q and R lie on a straight line - Q lies one third of t... show full transcript
Worked Solution & Example Answer:Relative to a fixed origin, points P, Q and R have position vectors p, q and r respectively - Edexcel - A-Level Maths Pure - Question 4 - 2020 - Paper 2
Step 1
P, Q and R lie on a straight line
Answer
Since points P, Q, and R are collinear, we can express the position vector of Q in relation to P and R. This means that Q can be represented as a linear combination of the vectors corresponding to P and R.
We can express this as: where is a scalar representing the position of Q on the line segment connecting P and R. Given that Q lies one third of the way from P to R, we have: . Thus,
Step 2
Q lies one third of the way from P to R
Answer
Given our previous expression for q, we will express it in a more detailed form:
Starting from our equation, we can manipulate it to isolate q on one side to compare with what we need to show:
Multiplying through by 3 to eliminate the fraction gives: Rearranging this gives: Now if we write r in terms of p and q, we have: Thus, we have shown the required result.