The angle between the lines $\hat{\mathbf{r}}=(2 \hat{\mathbf{i}}-3…
The angle between the lines $\hat{\mathbf{r}}=(2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}})+\lambda(\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \quad$ and $\hat{\mathbf{r}}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})$ is
$\frac{\pi}{2}$
$\cos ^{-1}\left(\frac{9}{\sqrt{91}}\right)$
$\cos ^{-1}\left(\frac{7}{\sqrt{84}}\right)$
$\frac{\pi}{2}$
Solution
Given lines are
and
$
\begin{aligned}
& \mathrm{r}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}+\lambda(\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \\
& \mathrm{r}=(\hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}})
\end{aligned}
$
Here DR's of given lines are $(1,4,3)$ and $(1,2,-3)$.
$\therefore$ Angle between these lines is
$
\begin{aligned}
\cos \theta & =\frac{a_1 a_2+b_1 b_2+c_1 c_2}{\sqrt{a_1^2+b_1^2+c_1^2} \sqrt{a_2^2+b_2^2+c_2^2}} \\
& =\frac{1 \times 1+4 \times 2+3 \times(-3)}{\sqrt{1^2+4^2+3^2} \sqrt{1^2+2^2+(-3)^2}} \\
& =\frac{1+8-9}{\sqrt{1+16+9} \sqrt{1+4+9}}=0 \\
\Rightarrow \quad \theta & =\frac{\pi}{2}
\end{aligned}
$