 # 2011 A Level H2 Math

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##### Suggested Handwritten and Video Solutions ##### Suggested Handwritten and Video Solutions      ##### 2011 A Level H2 Math Paper 1 Question 7 Referred to the origin $O$, the points $A$ and $B$ are such that $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$. The point $P$ on $OA$ is such that $OP:PA=1:2$, and the point $Q$ on $OB$ is such that $OQ:QB=3:2$. The mid-point of $PQ$ is $M$ (see diagram).

(i)

Find $\overrightarrow{OM}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ and show that the area of triangle $OPM$ can be written as $k\left| \mathbf{a}\times \mathbf{b} \right|$, where $k$ is a constant to be found.



(i) Find $\overrightarrow{OM}$ in terms of $\mathbf{a}$ and $\mathbf{b}$ and show that the area of triangle $OPM$ can be written as $k\left| \mathbf{a}\times \mathbf{b} \right|$, where $k$ is a constant to be found.



(ii)

The vectors $\mathbf{a}$ and $\mathbf{b}$ are now given by

$\mathbf{a}=2p\mathbf{i}-6p\mathbf{j}+3p\mathbf{k}$ and $\mathbf{b}=\mathbf{i}+\mathbf{j}-2\mathbf{k}$,

where $p$ is a positive constant. Given that $\mathbf{a}$ is a unit vector,

(ii) The vectors $\mathbf{a}$ and $\mathbf{b}$ are now given by

$\mathbf{a}=2p\mathbf{i}-6p\mathbf{j}+3p\mathbf{k}$ and $\mathbf{b}=\mathbf{i}+\mathbf{j}-2\mathbf{k}$,

where $p$ is a positive constant. Given that $\mathbf{a}$ is a unit vector,

(a) find the exact value of $p$,



(a) find the exact value of $p$,



(b) give a geometrical interpretation of $\left| \mathbf{a}\cdot \mathbf{b} \right|$,



(b) give a geometrical interpretation of $\left| \mathbf{a}\cdot \mathbf{b} \right|$,



(c) evaluate $\mathbf{a}\times \mathbf{b}$.



(c) evaluate $\mathbf{a}\times \mathbf{b}$.



##### Suggested Handwritten Solutions        ##### Suggested Handwritten and Video Solutions    ##### Suggested Handwritten and Video Solutions ##### Suggested Handwritten and Video Solutions        