2022 ACJC J2 MYE Q4

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Timothy Gan

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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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Published: 22nd February 2024
2022 ACJC J2 MYE Q4
Written by
Timothy Gan
This is Tim. Tim loves to teach math. Tim seeks to improve his teaching incessantly! Help Tim by telling him how he can do better.
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Published: 22nd February 2024

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