1. (a) 2x - y +z = 3
Direction ratios of the plane sitting club A is 2, −1, 1.
Hence equation of plane is 2x − y + z = 3
2. (b) \(\sqrt{14}\)
\(\vec n = \hat i + 3\hat j + 2\hat k\)
So, \(|\vec n| = \sqrt{1^2 +3^2 + 2^2}\)
\(= \sqrt{14}\)
3. (c) \(\frac x 8 + \frac y{8/3} + \frac z4 =1\)
Equation of plane & sitting club B is x + 3y + 2z = 8, it's intercept form is
\(\frac x 8 + \frac y{8/3} + \frac z{8/2} =1\) or \(\frac x 8 + \frac y{8/3} + \frac z4 =1\)
4. (d) Player sitting at (1, 1, 2)
Given equation is x + 3y + 2z = 8
or x − 1 + 3y − 3 + 2z − 4 = 0
or (x − 1) + 3(y − 1) + 2(z − 2) = 0
So, plane passing through (1, 1, 2).
5. (a) \(\frac 8{\sqrt{14}}\)
Distance = \(\left |\frac{-8}{\sqrt{1^2 + 3^2 + 2^2}}\right| \)
\(= \frac {8}{\sqrt{14}}\)