Show that the lines with direction cosines 12-13- -3-13- -4-13 - 4-13- 12-13- 3-13 - 3-13- -4-13- 12-13 are mutually perpendicular-

(i) For first two lines with direction cosines, 12/13, 3/13, 4/13 and 4/13, 12/13, 3/13 we obtain l_1 l_2 + m_1m_2 + n_1n_2 =12/13×4/13+( 3/13 )×12/13+( 4/13 ...

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Standard XII

Mathematics

General Solution of Trigonometric Equation

Show that the...

Question

Show that the lines with direction cosines $\frac{12}{13}, \frac{- 3}{13}, \frac{- 4}{13}; \frac{4}{13}, \frac{12}{13}, \frac{3}{13}; \frac{3}{13}, \frac{- 4}{13}, \frac{12}{13}$ are mutually perpendicular.

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Solution

(i) For first two lines with direction cosines, $\frac{12}{13}, \frac{- 3}{13}, \frac{- 4}{13} a n d \frac{4}{13}, \frac{12}{13}, \frac{3}{13}$ we obtain
$l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = \frac{12}{13} \times \frac{4}{13} + (\frac{- 3}{13}) \times \frac{12}{13} + (\frac{- 4}{13}) \times \frac{3}{13} = \frac{48}{169} - \frac{36}{169} - \frac{12}{169} = 0$
Therefore, the lines are perpendicular.
(ii) For second and third lines with direction cosines, $\frac{4}{13}, \frac{12}{13}, \frac{3}{13} a n d \frac{3}{13}, \frac{- 4}{13}, \frac{12}{13}$ we obtain
$l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = \frac{4}{13} \times \frac{3}{13} + \frac{12}{13} \times (\frac{- 4}{13}) + \frac{3}{13} \times \frac{12}{13} = \frac{12}{169} - \frac{48}{169} + \frac{36}{169} = 0$
Therefore, the lines are perpendicular.
(iii) For third and first lines with directions cosines, $\frac{3}{13}, \frac{- 4}{13}, \frac{12}{13} a n d \frac{12}{13}, \frac{- 3}{13}, \frac{- 4}{13}$ we obtain
$l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = \frac{3}{13} \times \frac{12}{13} + (\frac{- 4}{13}) \times (\frac{- 3}{13}) + \frac{12}{13} \times (\frac{- 4}{13}) = \frac{36}{169} + \frac{12}{169} - \frac{48}{169} = \frac{36}{169} - \frac{36}{169} = 0$
Therefore, the lines are perpendicular.
Hence, all the lines are mutually perpendicular.

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Show that the lines with direction cosines 1213, −313, −413; 413, 1213, 313; 313, −413, 1213 are mutually perpendicular.

Show that the lines with direction cosines $\frac{12}{13}, \frac{- 3}{13}, \frac{- 4}{13}; \frac{4}{13}, \frac{12}{13}, \frac{3}{13}; \frac{3}{13}, \frac{- 4}{13}, \frac{12}{13}$ are mutually perpendicular.