The direction cosines of the line x = y = z a | vedclass

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(A) The given line is $x = y = z$. This can be written as $\frac{x}{1} = \frac{y}{1} = \frac{z}{1}$.Comparing this with the symmetric form of a line $

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$\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}$

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(A) The given line is $x = y = z$. This can be written as $\frac{x}{1} = \frac{y}{1} = \frac{z}{1}$.Comparing this with the symmetric form of a line $\frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c}$,the direction ratios are $(a, b, c) = (1, 1, 1)$.The direction cosines $(l, m, n)$ are given by $l = \frac{a}{\sqrt{a^2+b^2+c^2}}$,$m = \frac{b}{\sqrt{a^2+b^2+c^2}}$,and $n = \frac{c}{\sqrt{a^2+b^2+c^2}}$.Here,$\sqrt{a^2+b^2+c^2} = \sqrt{1^2+1^2+1^2} = \sqrt{3}$.Thus,the direction cosines are $\left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right)$.

The direction cosines of the line $x = y = z$ are

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