The acute angle between the planes `2x-y+z=6\ a n d\ x+y+2z=3` is `45^0\ ` b. `60^0` c. `30^0` d. `75^0`

Explanation

We know, the angle between 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` is,

`theta=cos ^{-1} frac{{a}_{1} {a}_{2}+{b}_{1} {~b}_{2}+{c}_{1} {c}_{2}}{sqrt{{a}_{1}^{2}+{b}_{1}^{2}+{c}_{1}^{2}} cdot sqrt{{a}_{2}^{2}+{b}_{2}^{2}+{c}_{2}^{2}}}`

Hеге, `a_{1}=2, b_{1}=-1, c_{1}=1, d_{1}=-6` and `a_{2}=1, b_{2}=1, c_{2}=2`, `d_{2}=-3`


So, the acute angle between the planes `2 x-y+z=6` and `x+y+2 z=3` is


`theta=cos ^{-1} frac{((2 times 1)+((-1) times 1)+(1 times 2))}{sqrt{2^{2}+(-1)^{2}+1^{2}} cdot sqrt{1^{2}+1^{2}+2^{2}}}

=cos ^{-1} frac{(2-1+2)}{sqrt{4+1+1} cdot sqrt{1+1+4}}

=cos ^{-1} frac{3}{sqrt{6} cdot sqrt{6}}

=cos ^{-1} frac{3}{6}
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