With respect to the origin $$\(O\)$$, the points $$\(A\)$$ and $$\(B\)$$ have position vectors given by $$\(\overrightarrow{O A}=\left(\begin{array}{l}1 \\ 2 \\ 1\end{array}\right)\)$$ and $$\(\overrightarrow{O B}=\left(\begin{array}{r}3 \\ 1 \\ -2\end{array}\right)\)$$. The line $$\(l\)$$ has equation $$\(\mathbf{r}=\left(\begin{array}{l}2 \\ 3 \\ 1\end{array}\right)+\lambda\left(\begin{array}{r}1 \\ -2 \\ 1\end{array}\right)\)$$ Find the position vector of the point $$\(P\)$$ on $$\(l\)$$ such that $$\(A P=B P\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................
Exam No：9709_s21_qp_31 Year：2021 Question No：8(b)
Answer:
State or imply \(\pm \overrightarrow{A P}\) and \(\pm \overrightarrow{B P}\) in component form, i.e. \((1+\lambda, 1-2 \lambda, \lambda)\) and \((-1+\lambda, 2-2 \lambda, 3+\lambda)\), or equivalent
Form an equation in \(\lambda\) by equating moduli or by using \(\cos B A P=\cos A B P\)
Obtain a correct equation in any form \((1+\lambda)^{2}+(1-2 \lambda)^{2}+\lambda^{2}=(\lambda-1)^{2}+(2-2 \lambda)^{2}+(\lambda+3)^{2}\)
Solve for \(\lambda\) and obtain position vector
Obtain correct position vector for \(P\) in any form, e.g. \((8,-9,7)\) or \(8 \mathbf{i}-9 \mathbf{j}+7 \mathbf{k}\)
Form an equation in \(\lambda\) by equating moduli or by using \(\cos B A P=\cos A B P\)
Obtain a correct equation in any form \((1+\lambda)^{2}+(1-2 \lambda)^{2}+\lambda^{2}=(\lambda-1)^{2}+(2-2 \lambda)^{2}+(\lambda+3)^{2}\)
Solve for \(\lambda\) and obtain position vector
Obtain correct position vector for \(P\) in any form, e.g. \((8,-9,7)\) or \(8 \mathbf{i}-9 \mathbf{j}+7 \mathbf{k}\)
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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