To determine the direction cosines of the line given by the equations \(x = y = 1 - z\), we first need to understand the concept and steps involved:
- The given line equation can be interpreted as \(x = y = 1 - z\). This can be rewritten as two separate equations: \(x = 1 - z\) and \(y = 1 - z\).
- The direction ratios of the line are coefficients of \(x\), \(y\), and \(z\) in the vector form.
Here, by comparing equations, we note:- \(x = 1 - z \Rightarrow x + z = 1\). Thus, the coefficients of \(x\) and \(z\) are 1 and -1 respectively, making the direction ratios \((1, 1, -1)\).
- From the direction ratios \((1, 1, -1)\), calculate the direction cosines. Direction cosines \((l, m, n)\) are given by:
- \(l = \dfrac{a}{\sqrt{a^2 + b^2 + c^2}}\), \(m = \dfrac{b}{\sqrt{a^2 + b^2 + c^2}}\), \(n = \dfrac{c}{\sqrt{a^2 + b^2 + c^2}}\)
- \(l = \dfrac{1}{\sqrt{1^2 + 1^2 + (-1)^2}} = \dfrac{1}{\sqrt{3}}\)
- \(m = \dfrac{1}{\sqrt{3}}\)
- \(n = \dfrac{-1}{\sqrt{3}}\)
- Thus, the direction cosines of the line are \(\dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}, -\dfrac{1}{\sqrt{3}}\).
This matches the correct answer from the options provided:
\(\dfrac{1}{\sqrt{3}}, \dfrac{1}{\sqrt{3}}, -\dfrac{1}{\sqrt{3}}\).