Find the angle between the normal to the planes `2x-y+z=6\ a n d\ x+y+2z=7.`

Explanation

The given planes are `2x-y+z=6`and `x+y+2z=7.`

Let normal vector to plane first `vec (n_1)=2 hat(i)-hat(j)+hat(k)`

And normal vector to plane second `vec(n_2)=hat(i)+hat(j)+2hat(k)`

Then angle between `vec(n_1)` and `vec(n_2)`

`cos theta=(vec(n_1).vec(n_2))/(|vec(n_1)||vec(n_2)|)`

`cos theta= ((2hat(i)-hat(j)+hat(k)).(hat(i)+hat(j)+2hat(k)))/((sqrt(4+1+1))(sqrt(1+1+4)))`

`cos theta=|(2-1+2)/(sqrt6 .sqrt6)|`

`cos theta=1/2`

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