Referred to the origin, the points $A$ and $B$ have position vectors $\mathbf{a}$ and \(\mathb

Question

Referred to the origin, the points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ respectively. Given that $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors and $\left( \mathbf{a}+2\mathbf{b} \right)\cdot \left( \mathbf{a}-2\mathbf{b} \right)=0$, find $OA:OB$.[3] Points $C$ and $D$ have coordinates $\left( m,n,1 \right)$ and $\left( 1,2,2 \right)$ respectively, for some real numbers $m$ and $n$, such that $\overrightarrow{OC}$ is perpendicular to $\overrightarrow{OD}$.[2] Find a linear equation satisfied by $m$ and $n$.[2] Given further that triangle $OCD$ is an isosceles triangle, find the possible values of $m$ and $n$.[4]

Accepted Answer

(a) $OA:OB=2:1$ (b)(i) $m+2n+2=0$ (ii) When $n=\frac{2}{5}$, $m=-\frac{14}{5}$. When $n=-2$, $m=2$.

2023 TMJC Promo Q6

2023 TMJC Promo Q6

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