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Question:
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Consider the following statements:
$[2018-I I]$
1 . The angle between the planes
$2 x-y+z=1$ and $x+y+2 z=3$ is $\frac{\pi}{3}$
2 . The distance between the plancs
$6 x-3 y+6 z+2=0$ and $2 x-y+2 z+4=0$ is $\frac{10}{9}$
Which of the above statements is/are correct
Options:
$[2018-I I]$
1 . The angle between the planes
$2 x-y+z=1$ and $x+y+2 z=3$ is $\frac{\pi}{3}$
2 . The distance between the plancs
$6 x-3 y+6 z+2=0$ and $2 x-y+2 z+4=0$ is $\frac{10}{9}$
Which of the above statements is/are correct
Solution:
1235 Upvotes
Verified Answer
The correct answer is:
Both and 2
Angle between planes
$=\cos ^{-1}\left(\frac{2-1+2}{6}\right)$
$=\cos ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}$
Distance between planes
$=\frac{\left|\frac{2}{3}-4\right|}{\sqrt{9}}=\frac{10}{3 \times 3}=\frac{10}{9}$
$=\cos ^{-1}\left(\frac{2-1+2}{6}\right)$
$=\cos ^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{3}$
Distance between planes
$=\frac{\left|\frac{2}{3}-4\right|}{\sqrt{9}}=\frac{10}{3 \times 3}=\frac{10}{9}$
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