Relative to origin $O$, the position vectors of points $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b

Question

Relative to origin $O$, the position vectors of points $A$, $B$ and $C$ are $\mathbf{a}$, $\mathbf{b}$ and $\mathbf{c}$ respectively, where $\mathbf{b}$is a unit vector. It is given that $\mathbf{b}$ and $\mathbf{b}-\mathbf{a}$ are perpendicular and $C$ lies on $AB$ such that $AC:CB=3:1$. Show that $\left| \mathbf{a} ight|=\sec 	heta $, where $	heta $ is the angle between $\mathbf{a}$ and $\mathbf{b}$. [3] By expressing $\mathbf{c}$ in terms of $\mathbf{a}$ and $\mathbf{b}$, Find the value of $\left| \mathbf{c}\cdot \mathbf{b} ight|$ and state the geometrical interpretation of $\left| \mathbf{c}\cdot \mathbf{b} ight|$, [4] Find the value of $\frac{\left| \mathbf{b}	imes \mathbf{c} ight|}{\left| \mathbf{a}	imes \mathbf{b} ight|}$. [2]

Accepted Answer

(i) $|\mathbf{a}| = \sec\theta$ (ii) $|\mathbf{c} \cdot \hat{\mathbf{b}}| = 1$ (iii) $\frac{1}{4}$

2022 CJC Promo Q9

2022 CJC Promo Q9

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