If l_1, m_1, n_1; l_2, m_2, n_2 and l_3, m_3, | vedclass

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(B) Let the direction cosines of the required line be $(l, m, n)$. Since the line makes equal angles $	heta$ with the three mutually perpendicular li

Vedclass · Accepted Answer

$\frac{l_1 + l_2 + l_3}{\sqrt{3}}, \frac{m_1 + m_2 + m_3}{\sqrt{3}}, \frac{n_1 + n_2 + n_3}{\sqrt{3}}$

Vedclass · Answer

(B) Let the direction cosines of the required line be $(l, m, n)$. Since the line makes equal angles $	heta$ with the three mutually perpendicular lines,we have:$l \cdot l_1 + m \cdot m_1 + n \cdot n_1 = \cos 	heta$$l \cdot l_2 + m \cdot m_2 + n \cdot n_2 = \cos 	heta$$l \cdot l_3 + m \cdot m_3 + n \cdot n_3 = \cos 	heta$Squaring and adding these equations,we get:$(l \cdot l_1 + m \cdot m_1 + n \cdot n_1)^2 + (l \cdot l_2 + m \cdot m_2 + n \cdot n_2)^2 + (l \cdot l_3 + m \cdot m_3 + n \cdot n_3)^2 = 3 \cos^2 	heta$Using the property of direction cosines for mutually perpendicular lines,this simplifies to $l^2 + m^2 + n^2 = 3 \cos^2 	heta$. Since $l^2 + m^2 + n^2 = 1$,we have $3 \cos^2 	heta = 1$,so $\cos 	heta = \frac{1}{\sqrt{3}}$.Thus,$l = \frac{l_1 + l_2 + l_3}{\sqrt{3}}$,$m = \frac{m_1 + m_2 + m_3}{\sqrt{3}}$,$n = \frac{n_1 + n_2 + n_3}{\sqrt{3}}$.

If $l_1, m_1, n_1$; $l_2, m_2, n_2$ and $l_3, m_3, n_3$ are the direction cosines of three mutually perpendicular lines,find the direction cosines of a line that makes equal angles with these lines.

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