10.5#17

I used the formula cos(theta)=n1(dot)n2/|n1|*|n2|

where n1=(5,-2,3) and n2=(2,-5-5)

in the end I did theta=arccos(5/sqrt(2052))

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everything you did is just fine...except the problem was meant to fool you. Notice the second equation is written 2y - 5x -5z=2.  In order to get the normal vector you should rearrange it so it looks like -5x+2y-5z=2.  The correct normal vector is <-5,2,-5>.