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

Question:

Find the angle between the lines whose direction cosines are given by the equations \(l + m + n = 0\), \(l^2 + m^2 - n^2 = 0\).

Solution:

The given equations are

\(l + m + n = 0\) ......(i)

\(l^2 + m^2 - n^2 = 0\) .......(ii)

From equation (i) \(n = - (l + m)\)

Putting the value of \(n\) in equation (ii) we get

\(l^2 + m^2 + [ - (l + m)^2] = 0\)

⇒ \(l^2 + m^2 - l^2 - m^2 - 2lm = 0\)

⇒ \( - 2lm = 0\)

⇒ \(lm = 0\)

⇒ \((- m - n)m = 0\) .....[∵ \(l = - m - n\)]

⇒ \((m + n)m = 0\)

⇒ \(m = 0\) or \(m = - n\)

⇒ \(l = 0\) or \(l = - n\)

∴ Direction cosines of the two lines are

0, - n, n and - n, 0, n

⇒ 0, - 1, 1 and - 1, 0, 1

\(\therefore \cos \theta = \frac{{(0i - j + k) \cdot ( - i + 0j + k)}}{{\sqrt {{{(0)}^2} + {{( - 1)}^2} + {{(1)}^2}} \sqrt {{{( - 1)}^2} + {{(0)}^2} + {{(1)}^2}} }}\)

\(\Rightarrow \cos \theta = \frac{1}{{\sqrt 2 \sqrt 2 }}\)

\(\Rightarrow \theta = \frac{\pi }{3}\)

Thus, Required angle is \(\frac{\pi }{3}\)

Getting Info...

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