Exercise - 11.2 Show that the three lines ...

NCERT Class XII Mathematics Chapter - - Solutions

Question : 6 of 78

Marks: +1, -0

Exercise - 11.2

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.

Solution:

Let the lines, whose direction cosines are given, be

l_1, l_2

and

l_3

Let α be the angle between

l_1

and

l_2

, then

\cos \alpha = \left| \frac{1}{13} \cdot \frac{4}{13} + \left(-\frac{3}{13}\right) \cdot \frac{12}{13} + \left(-\frac{4}{13}\right) \cdot \frac{3}{13} \right|

= \frac{48-36-12}{169}=0

\Rightarrow \alpha = \frac{\pi}{2} \Rightarrow l_1 \perp l_2

Similarly, we can show that

l_2 \perp l_3

and

l_3 \perp l_1 .

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