Show that the three lines with direction cosines 1213-313-413-413-1213-313-313-413-1213 are mutually perpendicular-

Line 1: l_1=1213, m_1= 313, n_1= 413 Line 2: l_2=413, m_2=1213, n_2=313 l_1l_2+m_1m_2+n_1n_2=(1213×413)+( 313×1213)+( 413×313) ...

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Byju's Answer

Standard XII

Mathematics

Distance between Two Parallel Planes

Show that the...

Question

Show that the three 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

Line $1 :$ - $l_{1} = \frac{12}{13}, m_{1} = \frac{- 3}{13}, n_{1} = \frac{- 4}{13}$
Line $2 :$ - $l_{2} = \frac{4}{13}, m_{2} = \frac{12}{13}, n_{2} = \frac{3}{13}$
$l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = (\frac{12}{13} \times \frac{4}{13}) + (\frac{- 3}{13} \times \frac{12}{13}) + (\frac{- 4}{13} \times \frac{3}{13})$
$= \frac{48}{169} + \frac{- 36}{169} + \frac{- 12}{169} = \frac{48 - 48}{169}$
$= 0$
$∴ l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} = 0$
$∴$ Given Lines are perpendicular
Line $2 :$ - $l_{2} = \frac{4}{13}, m_{2} = \frac{12}{13}, n_{2} = \frac{3}{13}$
Line $3 :$ - $l_{3} = \frac{3}{13}, m_{3} = \frac{- 4}{13}, n_{3} = \frac{12}{13}$
$l_{3} l_{2} + m_{3} m_{2} + n_{3} n_{2} = (\frac{4}{13} \times \frac{3}{13}) + (\frac{12}{13} \times \frac{- 4}{13}) + (\frac{3}{13} \times \frac{12}{13})$
$= \frac{12}{169} + \frac{- 48}{169} + \frac{36}{169} = \frac{48 - 48}{169} = 0$
$∴ l_{3} l_{2} + m_{3} m_{2} + n_{3} n_{2} = 0$
$∴$ Two lines are perpendicular
Line $3 :$ - $l_{3} = \frac{3}{13}, m_{3} = \frac{- 4}{13}, n_{3} = \frac{12}{13}$
Line $1 :$ - $l_{1} = \frac{12}{13}, m_{1} = \frac{- 3}{13}, n_{1} = \frac{- 4}{13}$
$l_{1} l_{3} + m_{1} m_{3} + n_{1} n_{3} = (\frac{3}{13} \times \frac{12}{13}) + (\frac{- 4}{13} \times \frac{3}{13}) + (\frac{12}{13} \times \frac{- 4}{13})$
$= \frac{36}{169} + \frac{12}{169} + \frac{- 48}{169} = \frac{48 - 48}{169} = 0$
$∴ l_{1} l_{3} + m_{1} m_{3} + n_{1} n_{3} = 0$
$∴$ Two lines are perpendicular
Hence given three Lines are mutually perpendicular.

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Show that the three lines with direction cosines1213−313,−413;413,1213,313;313,−413,1213 are mutually perpendicular.

Show that the three 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.