Question
Under what condition are the two lines $$y = \frac{m}{\ell }x + \alpha ,\,z = \frac{n}{\ell }x + \beta \,;$$ and $$y = \frac{{m'}}{{\ell '}}x + \alpha ',\,z = \frac{{n'}}{{\ell '}}x + \beta '$$ orthogonal ?
A.
$$\alpha \alpha ' + \beta \beta ' + 1 = 0$$
B.
$$\left( {\alpha + \alpha '} \right) + \left( {\beta + \beta '} \right) = 0$$
C.
$$\ell \ell ' + mm' + nn' = 1$$
D.
$$\ell \ell ' + mm' + nn' = 0$$
Answer :
$$\ell \ell ' + mm' + nn' = 0$$
Solution :
Given two lines are :
$$\eqalign{
& y = \frac{m}{\ell }x + \alpha ,\,z = \frac{n}{\ell }x + \beta \,{\text{ and}} \cr
& y = \frac{{m'}}{{\ell '}}x + \alpha ',\,z = \frac{{n'}}{{\ell '}}x + \beta ' \cr} $$
These two lines can be represented as :
$$\eqalign{
& \frac{{y - \alpha }}{{\frac{m}{\ell }}} = \frac{{x - 0}}{1} = \frac{{z - \beta }}{{\frac{n}{\ell }}} \cr
& {\text{and }}\frac{{y - \alpha '}}{{\frac{{m'}}{{c'}}}} = \frac{{x - 0}}{1} = \frac{{z - \beta '}}{{\frac{{n'}}{{\ell '}}}} \cr} $$
They are orthogonal, if
$$\frac{m}{\ell } \times \frac{{m'}}{{\ell '}} + 1 \times 1 + \frac{n}{\ell } \times \frac{{n'}}{{\ell '}} = - 1 \Rightarrow \ell \ell ' + mm' + nn' = 0$$