With respect to the origin $$\(O\)$$, the points $$\(A\)$$ and $$\(B\)$$ have position vectors given by $$\(\overrightarrow{O A}=2 \mathbf{i}-\mathbf{j}\)$$ and $$\(\overrightarrow{O B}=\mathbf{j}-2 \mathbf{k}\)$$ The midpoint of $$\(A B\)$$ is $$\(M\)$$. The point $$\(P\)$$ on the line through $$\(O\)$$ and $$\(M\)$$ is such that $$\(P A: O A=\sqrt{7}: 1\)$$. Find the possible position vectors of $$\(P\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s21_qp_32 Year：2021 Question No：11(b)
Answer:
State or imply \(M\) has position vector \(\mathbf{i}-\mathbf{k}\)
Taking a general point of \(O M\) to have position vector \(\lambda \mathbf{i}-\lambda
\mathbf{k}\), express \(A P=\sqrt{7} O A\) as an equation in \(\lambda\)
State a correct equation in any form
Reduce to \(\lambda^{2}-2 \lambda-15=0\)
Solve a quadratic and state a position vector
Obtain answers \(5 \mathbf{i}-5 \mathbf{k}\) and \(-3 \mathbf{i}+3 \mathbf{k}\)
Alternative method for Question 11(b)
State or imply that \(O P=\gamma \sqrt{2}\)
State or imply that \(\cos \frac{1}{2} A O B=\sqrt{\frac{2}{5}}\) and use cosine rule to form an equation in \(\gamma\)
State a correct equation in any form
Reduce to \(\gamma^{2}-2 \gamma-15=0\)
Solve a quadratic and state a position vector
Obtain answers \(5 \mathbf{i}-5 \mathbf{k}\) and \(-3 \mathbf{i}+3 \mathbf{k}\)
Alternative method for Question 11(b)
State or imply \(M\) has position vector \(\mathbf{i}-\mathbf{k}\)
State or imply that \(A M=\sqrt{3}\)
Use Pythagoras to find \(M P\)
Obtain \(M P=4 \sqrt{2}\)
Correct method to find a position vector
Obtain answers \(5 \mathbf{i}-5 \mathbf{k}\) and \(-3 \mathbf{i}+3 \mathbf{k}\)
Taking a general point of \(O M\) to have position vector \(\lambda \mathbf{i}-\lambda
\mathbf{k}\), express \(A P=\sqrt{7} O A\) as an equation in \(\lambda\)
State a correct equation in any form
Reduce to \(\lambda^{2}-2 \lambda-15=0\)
Solve a quadratic and state a position vector
Obtain answers \(5 \mathbf{i}-5 \mathbf{k}\) and \(-3 \mathbf{i}+3 \mathbf{k}\)
Alternative method for Question 11(b)
State or imply that \(O P=\gamma \sqrt{2}\)
State or imply that \(\cos \frac{1}{2} A O B=\sqrt{\frac{2}{5}}\) and use cosine rule to form an equation in \(\gamma\)
State a correct equation in any form
Reduce to \(\gamma^{2}-2 \gamma-15=0\)
Solve a quadratic and state a position vector
Obtain answers \(5 \mathbf{i}-5 \mathbf{k}\) and \(-3 \mathbf{i}+3 \mathbf{k}\)
Alternative method for Question 11(b)
State or imply \(M\) has position vector \(\mathbf{i}-\mathbf{k}\)
State or imply that \(A M=\sqrt{3}\)
Use Pythagoras to find \(M P\)
Obtain \(M P=4 \sqrt{2}\)
Correct method to find a position vector
Obtain answers \(5 \mathbf{i}-5 \mathbf{k}\) and \(-3 \mathbf{i}+3 \mathbf{k}\)
Knowledge points:
3.7.3 calculate the magnitude of a vector, and use unit vectors, displacement vectors and position vectors (In 2 or 3 dimensions.)
3.7.4 understand the significance of all the symbols used when the equation of a straight line is expressed in the form r = a +tb, and find the equation of a line, given sufficient information
Solution:
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