The angle between the vectors $3\mathbf{i}-2\mathbf{j}$ and \(6\mathbf{i}+d\mathbf{j}-\sqrt{7}\mat

Question

The angle between the vectors $3\mathbf{i}-2\mathbf{j}$ and $6\mathbf{i}+d\mathbf{j}-\sqrt{7}\mathbf{k}$ is ${{\cos }^{-1}}\left( \frac{6}{13} \right)$. Show that $2{{d}^{2}}-117d+333=0$.[3] With reference to the origin $O$, the points $A$, $B$, $C$ and $D$ are such that $\overrightarrow{OA}=\mathbf{a}$, $\overrightarrow{OB}=\mathbf{b}$, $\overrightarrow{AC}=5\mathbf{a}$ and $\overrightarrow{BD}=3\mathbf{b}$. The lines $AD$ and $BC$ cross at $E$ (see diagram). Find $\overrightarrow{OE}$ in terms of $\mathbf{a}$ and $\mathbf{b}$.[6] The point $F$ divides the line $CD$ in the ratio $5:3$. Show that $O$, $E$ and $F$ are collinear, and find $OE:OF$.[4]

Accepted Answer

(a) $2d^2 - 117d + 333 = 0$; (b)(i) $\overrightarrow{OE} = \frac{18}{23}\mathbf{a} + \frac{20}{23}\mathbf{b}$; (ii) $OE:OF = 8:23$

N2017 Specimen Paper P2 Q3

N2017 Specimen Paper P2 Q3

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