lf $a_{1}, b_{1}, c_{1}$ and $a_{2}, b_{2}, c_{2}$ are the $d . r . s$ of two lines then $(a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2})^{2} + Σ (b_{I} c_{2} - b_{2^{C_{1}}})^{2} =$

(a_{1} + b_{1} + c_{1}) (a_{2} + b_{2} + c_{2})

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(a_{1} b_{1} c_{1}) (a_{2} b_{2^{C} 2})

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1

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(a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) (a_{2}^{2} + b_{2}^{2} + c_{2}^{2})

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Solution

The correct option is D $(a_{1}^{2} + b_{1}^{2} + c_{1}^{2}) (a_{2}^{2} + b_{2}^{2} + c_{2}^{2})$
Given $a_{1}, b_{1}, c_{1}$ and $a_{2}, b_{2}, c_{2}$ are $d . r . s$ of two lines
Vector equation of lines with $a_{1}, b_{1}, c_{1}$ as $d . r . s$ is
$a_{1}^i + b_{1}^j + c_{1}^k$
Vector equation of line with $a_{2}, b_{2}, c_{2}$ as $d . r . s$ is
$a_{2}^i + b_{2}^j + c_{2}^k$
Let $θ$ be the angle between two lines
Formula: Dot product $A \cdot B = | A | | B | cos θ$
Cross product: $| A \times B | = | A | | B | sin θ$
Dot product of two lines
$a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = (\sqrt{{a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}}) (\sqrt{{a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2}}) cos θ$
Squaring on both sides
${(a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2})}^{2} = ({a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}) ({a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2}) {cos}^{2} θ$
${cos}^{2} θ = \frac{{(a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2})}^{2}}{({a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}) ({a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2})} . . . (1)$
Cross Product
$A \times B = ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k a_{1} & b_{1} & c_{1} a_{2} & b_{2} & c_{2} \end{matrix} ∣ ∣ ∣ ∣ ∣ A \times B =^i (b_{1} c_{2} - b_{2} c_{1}) +^j (c_{1} a_{2} - c_{2} a_{1}) +^k (a_{1} b_{2} - a_{2} b_{1}) | A \times B | = \sqrt{{(b_{1} c_{2} - b_{2} c_{1})}^{2} + {(c_{1} a_{2} - c_{2} a_{1})}^{2} + {(a_{1} b_{2} - a_{2} b_{1})}^{2}} | A \times B | = \sqrt{\sum {(b_{1} c_{2} - b_{2} c_{1})}^{2}}$

By cross product formula,
$\sqrt{\sum {(b_{1} c_{2} - b_{2} c_{1})}^{2}} = (\sqrt{{a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}}) (\sqrt{{a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2}}) sin θ$
Squaring on both sides,
$\sum {(b_{1} c_{2} - b_{2} c_{1})}^{2} = ({a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}) ({a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2}) {sin}^{2} θ {sin}^{2} θ = \frac{\sum {(b_{1} c_{2} - b_{2} c_{1})}^{2}}{({a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}) ({a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2})} . . . . (2)$

We know that
${sin}^{2} θ + {cos}^{2} θ = 1 . . . . (3)$
Adding $(1), (2)$
$\frac{{(a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2})}^{2}}{({a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}) ({a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2})} + \frac{\sum {(b_{1} c_{2} - b_{2} c_{1})}^{2}}{({a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}) ({a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2})} = {sin}^{2} θ + {cos}^{2} θ$

From $(3)$ ,
${(a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2})}^{2} + \sum {(b_{1} c_{2} - b_{2} c_{1})}^{2} = ({a_{1}}^{2} + {b_{1}}^{2} + {c_{1}}^{2}) ({a_{2}}^{2} + {b_{2}}^{2} + {c_{2}}^{2})$

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