The line ${{l}_{1}}$ has equation \(\mathbf{r}=\left( 3\mathbf{i}+2\mathbf{j}+\mathbf{k} \right)+\

Question

The line ${{l}_{1}}$ has equation $\mathbf{r}=\left( 3\mathbf{i}+2\mathbf{j}+\mathbf{k} \right)+\lambda \left( \mathbf{i}+\mathbf{j}+\mathbf{k} \right),\lambda \in \mathbb{R}$, and the line ${{l}_{2}}$ passes through the origin $O$ and point $A$, whose position vector is given by $\mathbf{a}=2\mathbf{i}+\mathbf{j}$. Find the acute angle between ${{l}_{1}}$ and ${{l}_{2}}$,[2] ${{l}_{1}}$ and the $xy$-plane.[2] A point $B$ on ${{l}_{1}}$ is such that $OB=\sqrt{5}$. Find the possible position vectors of $B$.[4]

Accepted Answer

(i) $39.2^\circ$ (ii) $35.3^\circ$ (iii) $\overrightarrow{OB}=\begin{pmatrix}2\1\0\end{pmatrix}$ or $\begin{pmatrix}0\-1\-2\end{pmatrix}$

2015 RI P1 Q6

2015 RI P1 Q6

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