Relative to an origin $O$, points $C$ and $D$ have position vectors $7\mathbf{i}+3\mathbf{j}+2\mathb

Question

Relative to an origin $O$, points $C$ and $D$ have position vectors $7\mathbf{i}+3\mathbf{j}+2\mathbf{k}$ and $10\mathbf{i}+a\mathbf{j}+b\mathbf{k}$ respectively, where $a$ and $b$ are constants. The straight line through $C$ and $D$ has equation $\mathbf{r}=\left( \begin{matrix} 7 \ 3 \ 2 \ \end{matrix} ight)+t\left( \begin{matrix} 1 \ 3 \ 0 \ \end{matrix} ight)$, $t\in \mathbb{R}$. Find the values of $a$ and $b$. Calculate the acute angle between the straight line though $C$ and $D$ and the $y$-axis, giving your correct answer to the nearest ${{0.1}^{\circ }}$. Find the position vector of the point $Q$ on the line $CD$ such that the angle between $\overrightarrow{OQ}$ and $\overrightarrow{OC}$ is equal to the angle between $\overrightarrow{OQ}$ and $\overrightarrow{OD}$ .

Accepted Answer

(i) a = 12, b = 2; (ii) 18.4°; (iii) OQ = (8, 6, 2)

N1995 P1 Q12 [Modified]

N1995 P1 Q12 [Modified]

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