Question

Find the direction cosines of a line that pass through the points $P(1,4,6)$ and $Q(5,1,11)$ and is so directed that it makes an acute angle with the positive direction of $y-$axis.

Answer 1

$\begin{array}{c}\overrightarrow{p}=\left(1,4,6\right)\Rightarrow \hat{i}+4\hat{j}+6\hat{k}\\ \overrightarrow{Q}=\left(5,1,11\right)\Rightarrow 5\hat{i}+\hat{j}+11\hat{k}\\ P\overrightarrow{Q}=\overrightarrow{Q}-\overrightarrow{P}\\ \Rightarrow \left(5\hat{i}+\hat{j}+11\hat{k}\right)-\left(\hat{i}+4\hat{j}+6\hat{k}\right)\\ \overrightarrow{PQ}=5\hat{i}+\hat{j}+11\hat{k}-\hat{i}-4\hat{j}+6\hat{k}\\ \overrightarrow{PQ}=4\hat{i}-3\hat{j}+5\hat{k}\\ DR=\left(4,-3,5\right)\\ D^{\prime }C=\frac{4}{\sqrt{4^2+{\left(-3\right)}^2+5^2}},\frac{-3}{\sqrt{4^2+{\left(-3\right)}^2+{\left(5\right)}^2}},\frac{5}{\sqrt{{\left(4\right)}^2+{\left(-3\right)}^2+{\left(5\right)}^2}}\\ \Rightarrow \frac{4}{\sqrt{50}},\frac{-3}{\sqrt{50}},\frac{5}{\sqrt{50}}\\ ⟨L,M,N⟩\Rightarrow ⟨\frac{4}{5\sqrt{2}},\frac{-3}{5\sqrt{3}},\frac{5}{5\sqrt{2}}⟩\end{array}$