Final answer:
The vectors u = <-3,9,6> and v = <4, -12, -8> are parallel because their corresponding components lead to the same scalar when divided, although they are not orthogonal since their dot product is not zero.
Step-by-step explanation:
To determine if the given vectors u = <-3,9,6> and v = <4, -12, -8> are orthogonal, parallel, or neither, we can use the dot product of the vectors. The dot product of two vectors is calculated by multiplying their corresponding components and then adding those products together. For vectors u and v, the dot product is:
u · v = (-3)*(4) + (9)*(-12) + (6)*(-8) = -12 - 108 - 48 = -168.
Since the dot product is not zero, the vectors are not orthogonal. To check if they are parallel, we look for a scalar k such that v = ku. Dividing the corresponding components of v by those of u, we get:
4 / -3 = -4/3,
-12 / 9 = -4/3,
-8 / 6 = -4/3.
All components lead to the same scalar, k = -4/3, which means that u and v are indeed parallel.