Assertion :The direction cosines of one of the angular bisectors of two intersecting lines having direction cosines as $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ are proportional to $l_{1} + l_{2}, m_{1} + m_{2}, n_{1} + n_{2}$ . Reason: The angle between the two intersecting lines having direction cosines as $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ is given by $cos θ = l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}$ .

Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

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Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

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Assertion is correct but Reason is incorrect

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Assertion is incorrect but Reason is correct

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Solution

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
We know that vector in the direction of angular bisector of unit vectors $\to a$ and $\to b$ is $\frac{\to a + \to b}{2 cos (θ / 2)}$
where $\to a = \to A B = l_{1} ˆ i + m_{1} ˆ j + n_{1} ˆ k$
and $\to b = \to A D = l_{2} ˆ i + m_{2} ˆ j + n_{3} ˆ k$

Thus, unit vector along the bisector is
$\frac{l_{1} + l_{2}}{cos (θ / 2)} ˆ i + \frac{m_{1} + m_{2}}{cos (θ / 2)} ˆ j + \frac{n_{2} + n_{2}}{cos (θ / 2)} ˆ k$

Hence, Statement $1$ is true.

Also, in triangle $A B D$ , by cosine rule

$cos θ = \frac{A B^{2} + A D^{2} - B D^{2}}{2 A B \cdot A D}$
$\frac{1 + 1 - {∣ ∣ (l_{1} - l_{2}) ˆ i + (m_{1} - m_{2}) ˆ j + (n_{1} - n_{2}) ˆ k ∣ ∣}_{1}^{2}}{2}$
$= \frac{2 - [(l_{1} - l_{2})^{2} + (m_{1} + m_{2})^{2} + (n_{1} - n_{2})^{2}]}{2}$
$= \frac{2 - [2 - 2 (l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2})]}{2}$
$= l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2}$

Hence, Statement 2 is true but not explain Statement $1$ .

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Similar questions

If two lines have direction cosines

$l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$

then the condition for these lines being perpendicular to each other would be -

l_{1}, m_{1}, n_{1}

l_{2}, m_{2}, n_{2}

l_{1}, m_{1}, n_{1}

l_{2}, m_{2}, n_{2}

l_{1}, m_{1}, n_{1}; l_{2}, m_{2}, n_{2}; l_{3}, m_{3}, n_{3}

∣ ∣ ∣ ∣ \begin{matrix} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} l_{3} & m_{3} & n_{3} \end{matrix} ∣ ∣ ∣ ∣ = 0

\frac{x - x_{1}}{l_{1}} = \frac{y - y_{1}}{m_{1}} = \frac{z - z_{1}}{n_{1}}, \frac{x - x_{2}}{l_{2}} = \frac{y - y_{2}}{m_{2}} = \frac{z - z_{2}}{n_{2}}

∣ ∣ ∣ ∣ \begin{matrix} x - x_{1} & y - y_{1} & z - z_{1} l_{1} & m_{1} & n_{1} l_{2} & m_{2} & n_{2} \end{matrix} ∣ ∣ ∣ ∣ = 0

(l_{1}, m_{1}, n_{1})

(l_{2}, m_{2}, n_{2})

(l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2})^{2} + \sum (m_{1} n_{2} - m_{2} n_{1})^{2}

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