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 -

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Both A and B

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None of these

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Solution

The correct option is C Both A and B
Two lines being parallel to each other means the angle between these lines is $0^{\circ}$ We know how to calculate the angle between two lines.
i.e.cos $(θ)$ = $(l_{1} . l_{2} + m_{1} . m_{2} + n_{1} . n_{2})$
Where $l_{1}, m_{1}, n_{1}$ are the direction cosines of one line and $l_{2}, m_{2}, n_{2}$ are the direction cosines of another.
If we put $θ$ = $0^{\circ} o r 180^{\circ}$ , we’ll get the condition for parallelism
So, $c o s (0^{\circ}) = (l_{1} . l_{2} + m_{1} . m_{2} + n_{1} . n_{2})$
Or, $l_{1} . l_{2} + m_{1} . m_{2} + n_{1} . n_{2} = 1$
And $c o s (180^{\circ}) = (l_{1} . l_{2} + m_{1} . m_{2} + n_{1} . n_{2})$
$l_{1} . l_{2} + m_{1} . m_{2} + n_{1} . n_{2} = - 1$
So, Option A is correct.
We also know that direction cosines are unique to a line.
So let’s say I have direction cosines of a line in vector form (l.i + m.j + n.k) . If I multiply it by a scalar quantity K, we’ll get K. l i + K.m j + K.n k
Which is also parallel to the previous line as the direction cosines doesn’t change.
So the direction cosines for new line will be -
$\frac{K . l}{\sqrt{K^{2} (l^{2} + m^{2} + n^{2})}}, \frac{K . m}{\sqrt{K^{2} (l^{2} + m^{2} + n^{2})}}, \frac{K . n}{\sqrt{K^{2} (l^{2} + m^{2} + n^{2})}}$
Which will be equal to l , m , n as $(l^{2} + m^{2} + n^{2} = 1)$
So, for parallel lines we can say that respective ratios of direction cosines are =\pm1\)
$\frac{l_{1}}{l_{2}} = \frac{m_{1}}{m_{2}} = \frac{n_{1}}{n_{2}} = \pm 1$

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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 -

Q. 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 -

(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}}

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}

If three mutually perpendicular lines have direction cosines $(l_{1}, m_{1}, n_{1}), (l_{2}, m_{2}, n_{2})$ and $(l_{3}, m_{3}, n_{3})$ line having direction cosines $l_{1} + l_{2} + l_{3}, m_{1} + m_{2} + m_{3}$ and $n_{1} + n_{2} + n_{3}$ make an
angle of..... with each other

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If two lines have direction cosines l1,m1,n1 and l2,m2,n2 then the condition for these lines being parallel to each other would be -

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 -