Question

$(l_1,m_1,n_1)$ and $(l_2,m_2,n_2)$ are d.c's two lines. Then
The lines are perpendicular if $l_1{l}_2+m_1{m}_2+n_1{n}_2=0$

Answer 1

Let $L_1$ has d.cs $l_1,m_1,n_1$

$\Rightarrow l_1$ is parallel to $l_1\hat{i}+m_1\hat{j}+n_{1}k$

Let $L_2$ has d.cs $l_2,m_2,n_2$

$\Rightarrow l_2$ is parallel to $l_2,m_2,n_2$

$\Rightarrow l_2$ is parallel to $l_2\hat{i}+m_2\hat{j}+n_2\hat{k}$

If $\overrightarrow{L_1}$ $\perp$ $\overrightarrow{L_2}$

$\Rightarrow \overrightarrow{L_1}\cdot \overrightarrow{L_2}=0$

$\Rightarrow (l_1\hat{i}+m_1\hat{j}+n_1\hat{k})\cdot (l_2\hat{i}+m_2\hat{j}+n_2\hat{k})$

$\Rightarrow l_1{l}_2+m_1{m}_2+n_1{n}_2=0$.