1. (d) (1, -1, 1)
Directions are coefficients of x, y, z are (1, −1, 1).
2. (a) \(\frac 2{\sqrt 3}\) units
\(\left|\frac{1 - 0+ 1-4}{\sqrt{1^2 + 1^2 + 1^2}}\right| = \frac 2{\sqrt 3}\)
3. (c) \(\frac {x -1}1 = \frac{y - 0}{-1} = \frac {z-1}1\)
\(\frac {x - x_1}a = \frac{y - y_1}b = \frac{z - z_1}c\)
where (x1, y1, z1) = (1, 0, 1)
(a, b, c) ≡ (1, -1, 1)
⇒ \(\frac {x -1}1 = \frac{y - 0}{-1} = \frac {z-1}1\)
4. (d) Both (a) and (c)
x − y + z + (2 − √3) and x − y + z + (2 + √3) both are correct equations.
5. (b) \(\left(\frac 1{\sqrt 3}, - \frac 1{\sqrt 3}, \frac 1{\sqrt 3}\right)\)
Given equation is x − y + z − 4 = 0
Normal form of equation is \(\frac 1{\sqrt 3}x - \frac 1{\sqrt 3}y + \frac 1{\sqrt 3}z = \frac 4{\sqrt 3}\)
Hence directions are \(\frac 1{\sqrt 3}, - \frac 1{\sqrt 3}, \frac 1{\sqrt 3}\).