Question

The angle between the lines 3x + 2y + z = 0 = x + y – 2z and 2x – y – z = 0 = 7x + 10y – 8z is :

• A. $\frac{\pi }{6}$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{3}$
• D. $0$

Answer 1

The correct option is B $\frac{\pi }{2}$

Let $l_1,m_1,n_1$ be D.R.’s of the line
x + y – 2z = 0 = 3x + 2y + z
$\Rightarrow l_1+m_1-2n_1=0$
$\Rightarrow 3l_1+2m_1+n_1=0\Rightarrow \frac{l_1}{5}=\frac{m_1}{-7}=\frac{n_1}{-1}.....\left(1\right)$
Let $l_2,m_2,n_2$ be D.R.’s of the line
2x – y – z = 0 = 7x + 10y – 8z
$\Rightarrow 2l_2-m_2-n_2=0$
$7l_2+10m_2-8n_2=0$
$\Rightarrow \frac{l_2}{18}=\frac{m_2}{9}=\frac{n_2}{27}$
i.e., $\frac{l_2}{2}=\frac{m_2}{1}=\frac{n_2}{3}....\left(2\right)$
Let $\theta$ be angle between two lines. Then
$cos\theta =\frac{10-7-3}{\sqrt{4+1+9}\sqrt{25+49+1}}=0\Rightarrow \theta =\frac{\pi }{2}$