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Home » Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l2 + m2 – n2 = 0.
Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l2 + m2 – n2 = 0.
CBSE | Class 12 | Exercise 1 | Maths | NCERT Exemplar | Three Dimensional Geometry
Find the angle between the lines whose direction cosines are given by the equations l + m + n = 0, l2 + m2 – n2 = 0.

According to ques,

Equations are,

    \[l~+~m~+~n~=\text{ }0\text{ }\ldots ..\text{ }\left( i \right)\]

    \[{{l}^{2}}~+~{{m}^{2}}~~{{n}^{2}}~=\text{ }0\text{ }\ldots .\text{ }\left( ii \right)\]

From equation (i),

    \[~n\text{ }=\text{ }\text{ }\left( l\text{ }+\text{ }m \right)\]

Putting the value of n is equation (ii),

    \[{{l}^{2}}~+\text{ }{{m}^{2}}~+\text{ }{{\left[ -\left( l\text{ }+\text{ }m \right) \right]}^{2}}~=\text{ }0\]

Or,

    \[{{l}^{2}}~+\text{ }{{m}^{2}}~\text{ }{{l}^{2}}~\text{ }{{m}^{2}}~\text{ }2lm\text{ }=\text{ }0\]

Or,

    \[-2lm\text{ }=\text{ }0\]

Or,

    \[lm\text{ }=\text{ }0\Rightarrow \left( -\text{ }m\text{ }\text{ }n \right)m\text{ }=\text{ }0\text{ }\left[ Since,\text{ }l\text{ }=\text{ }\text{ }m\text{ }\text{ }n \right]\]

Or,

    \[\left( m\text{ }+\text{ }n \right)m\text{ }=\text{ }0\Rightarrow m\text{ }=\text{ }0\text{ }or\text{ }m\text{ }=\text{ }-n\]

    \[\Rightarrow l\text{ }=\text{ }0\text{ }or\text{ }l\text{ }=\text{ }-n\]

Direction cosines of the two lines are:

    \[0,\text{ }-n,\text{ }n\text{ }and\text{ }-n,\text{ }0,\text{ }n\Rightarrow 0,\text{ }-1,\text{ }1\text{ }and\text{ }-1,\text{ }0,\text{ }1\]

Hence, the required angle is

    \[\pi /3.\]

i

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