If two lines with direction cosines l 1- m 1- n 1 and l 2- m 2- n 2 intersect- then the angle between them would be -A- cos -1l1- l2-m1- m2-n1- n2B- tan -1l1- l2-m1- m2-n1- n2C- -1l1- l2-m1- m2-n1- n2D- sin -1l1- l2-m1- m2-n1- n2

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If two lines ...

Question

If two lines with direction cosines $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ intersect, then the angle between them would be -

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Solution

The correct option is B
Let’s try to understand this problem with the help of vectors. Let there be two vectors a and b with direction cosines $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ respectively. Then, $a = l_{1} i + m_{1} j + n_{1} k$ and $b = l_{2} i + m_{2} j + n_{2} k$

$c o s (θ) = \frac{a . b}{| a | | b |}$
$c o s (θ) = \frac{(l_{1} i + m_{1} j + n_{1} k) . (l_{2} i + m_{2} j + n_{2} k)}{| l_{1} i + m_{1} j + n_{1} k | | l_{2} i + m_{2} j + n_{2} k |}$
$c o s (θ) = \frac{(l_{1} . l_{2} + m_{1} . m_{2} + n_{1} . n_{2})}{\sqrt{l_{1}^{2} + m_{1}^{2} + n_{1}^{2}} \sqrt{l_{2}^{2} + m_{2}^{2} + n_{2}^{2}}}$
$c o s (θ) = l_{1} l_{2} + m_{1} m_{2} + n_{1} . n_{2}$ (since $l_{1}^{2} + m_{1}^{2} + n_{1}^{2}; l_{2}^{2} + m_{2}^{2} + m_{2}^{2} + n_{2}^{2} = 1$
Or, $θ = c o s^{- 1} (l_{1} . l_{2} + m_{1} . m_{2} + n_{1} . n_{2})$
These two vectors a and b with direction cosines $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ can be used to represent lines with direction cosines $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ So the angle between them can be given by $c o s^{- 1}$ $(l_{1} . l_{2} + m_{1} . m_{2} + n_{1} . n_{2})$

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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 -

Q. 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 parallel to each other would be -

Q. 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}

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

and

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

are d.c's two lines. Then
The lines are parallel is

\frac{l_{1}}{l_{2}} = \frac{m_{1}}{m_{2}} = \frac{n_{1}}{n_{2}}

If $l_{1}, m_{1}, n_{1}, l_{2}, m_{2}, n_{2} a n d l_{3}, m_{3}, n_{3}$ are the direction cosines of three mutually perpendicular lines,then prove that the line whose direction cosines are proportional to $l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3} a n d n_{1} + n_{2} + n_{3}$ makes equal angles with them.

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If two lines with direction cosines l1,m1,n1 and l2,m2,n2 intersect, then the angle between them would be -

If two lines with direction cosines $l_{1}, m_{1}, n_{1}$ and $l_{2}, m_{2}, n_{2}$ intersect, then the angle between them would be -