10.4#5

I need help on this problem on how to find the orthogonal vectors
of 'a' and 'b'. I've tried it many times! I computed the cross product of
'a' and 'b' and then used the dot product using each vectors. Please
explain to me what I'm doing wrong.

(you can click on the image to enlarge it)

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Well, you're doing a number of things wrong, starting with your interpretation of the problem.
First of all, your supposed to find two unit vectors u^ and v^, each of which is orthogonal to both of the vectors a and b.   One thing you choose to do is to compute the cross product a x b, because this will give you a vector orthogonal to both a and b.  It looks like you made a simple arithmetic error in this computation however, because for example
 a x b = (-3i-5j+4k)x (-2i+4j+4k)
         = -12( i x j)+10( j x i)-12( i x k)-8( k x i)-20( j x k)+16( k x j)
         = -22( i x j) - 4( i x k) - 36( j x k)
         = -22 k + 4 j - 36 i
         = - 36 i + 4 j - 22 k

While this vector is orthogonal to both a and b,  it is not a unit vector.  Since the norm of this vector is ||a x b||=√(36^2+4^2+22^2)=√1796=2√449, the unit vector pointing in the direction of  a x b is
 a x b/(2√449). Of course another unit vector perpendicular to both  a and b is b x a/(2√449), which points in the opposite direction