Question

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

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{2}{3}$
• D. None of the above

Answer 1

The correct option is B $\frac{1}{3}$

The given planes are $x+y+x+I=0$ and $2x-2y+2z+I=0$

For two planes, $a_{1}x+b_{1}y+c_{1}z+d_1=0$ and $a_{2}x+b_{2}y+c_{2}z+d_2=0$ the cosine of the angle between them is,

$\cos \theta =\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}$

So, for the given planes we have

$\cos \theta =\frac{1\times 2+1\times (-2)+1\times 2}{\sqrt{3}\sqrt{12}}=\frac{2}{6}=\frac{1}{3}$

Hence, option B is correct.