Question

Find the angle between the lines whose direction cosines are $<\frac{\sqrt{3}}{2},\frac{1}{4},\frac{\sqrt{3}}{4}>and<-\frac{\sqrt{3}}{2},\frac{1}{4},\frac{\sqrt{3}}{4}>.$

Answer 1

Direction Cosines are $<\frac{\sqrt{3}}{2},\frac{1}{4},\frac{\sqrt{3}}{4}>$ and $<\frac{-\sqrt{3}}{2},\frac{1}{4},\frac{\sqrt{3}}{4}>$

$\Rightarrow$ Let $Q$ is the angle between lines with Direction Cosines $<l_1,m_1,n_1>$ and $<l_2,m_2.n_2>$

$\Rightarrow \cos 2=l_1{l}_2+m_1{m}_2+n_1{n}_2$

$=\left(\frac{-\sqrt{3}}{2}\right)\frac{\sqrt{3}}{2}+\frac{1}{4}\times \frac{1}{4}+\left(\frac{\sqrt{3}}{4}\right)\frac{\sqrt{3}}{4}$

$=\frac{-3}{4}+\frac{1}{16}+\frac{3}{16}=-\frac{3}{4}+\frac{4}{16}$

$=\frac{-3}{4}+\frac{1}{4}=-\frac{2}{4}$

$\Rightarrow \cos 2=-\frac{1}{2}\Rightarrow Q={120}^o$