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

1. 1/2
2. 1/3
3. 2/3
4. None of the above

Option 2 : 1/3
Free
Electric charges and coulomb's law (Basic)
99 K Users
10 Questions 10 Marks 10 Mins

## Detailed Solution

Concept:

The angle between the two planes a1x + b1y + c1z + d= 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 + d= 0

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

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

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

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

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

$$\rm cos\;\theta = \frac {2}{\sqrt {3}\times2\sqrt{3}}$$

⇒ $$\rm\cos\;\theta = \frac{1}{3}$$
{"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0}