Find a vector orthogonal to both <−5,3,0> and to <0,3,−3> of the form

<1,___,___>

Just take the cross product:

< −5, 3, 0 > x < 0, 3, −3 > = < −9, −15, −15 > = −9 < 1, 5/3, 5/3 >

Answer: < 1, 5/3, 5/3 >

u = -5i + 3j

v = 3j - 3k

The vector orthogonal to both u and v is u x w

i.........j.........k
-5........3........0
0........3........-3

= i(-9 - 0) - j(15 - 0) + k(-15 - 0)

= -9i - 15j - 15k

= -3i - 5j - 5k