What is the cosine of angle between the planes x + y + z + 1 = 0 and 2x - 2y + 2z + 1 = 0?

Concept:

 The angle between the two planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is given by 

\(\rm cos\;\theta = \frac{a_1a_2+ b_1b_2+c_1c_2}{\sqrt {(a_1^2+b_1^2+c_1^2)}\times\sqrt{(a_2^2+b_2^2+c_2^2)}}\)

Calculations:

Given planes are,  x + y + z + 1 = 0 ⇒ a1x + b1y + c1z + d1 = 0

and 2x - 2y + 2z + 1 = 0 ⇒ a2x + b2y + c2z + d2 = 0 

We know that the angle between the two planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 is given by 

\(\rm cos\;\theta = \frac{a_1a_2+ b_1b_2+c_1c_2}{\sqrt {(a_1^2+b_1^2+c_1^2)}\times\sqrt{(a_2^2+b_2^2+c_2^2)}}\)

⇒ \(\rm cos\;\theta = \frac {(1)(2)+(1)(-2)+(1)(2)}{\sqrt {1^2+1^2+1^2}\times\sqrt{(2^2+(-2)^2+2^2)}}\)

⇒\(\rm cos\;\theta = \frac {(2)+(-2)+(2)}{\sqrt {3}\times\sqrt{12}}\)

⇒\(\rm cos\;\theta = \frac {2}{\sqrt {3}\times2\sqrt{3}}\)

⇒ \(\rm\cos\;\theta = \frac{1}{3}\)