**2022 ACJC J2 MYE Q4 **

(a)

Vectors $\mathbf{u}$, $\mathbf{v}$ and $\mathbf{w}$ are such that $\mathbf{w}\ne \mathbf{0}$ and $4\mathbf{w}\times \mathbf{v}=3\mathbf{u}\times \mathbf{w}$.

(i) Show that $\mathbf{w}=\lambda \left( 3\mathbf{u}+4\mathbf{v} \right)$, where $\lambda $ is a constant.

[2]

(ii) It is now given that $\mathbf{u}$ and $\mathbf{v}$ are unit vectors which are perpendicular. Use a suitable scalar product to find the modulus of vector $\mathbf{w}$ in terms of $\lambda $.

[2]

(b)

Referred to the origin $O$, points $A$ and $B$ and $C$ have position vectors $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively. Point $C$ lies on line $AB$ produced such that $AC:AB=k:1$, where $k$ is a constant and $1<k<2$. Show that $\mathbf{c}=\left( 1-k \right)\mathbf{a}+k\mathbf{b}$.

[1]

$D$ is the mid-point of $OA$. The lines $DB$ produced and $OC$ produced meet at the point $E$. Given that $k=\frac{3}{2}$, find, in terms of $\mathbf{a}$ and $\mathbf{b}$, the position vector of $E$ and find the ratio $DB:DE$.

[4]

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