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Answer: c
Explanation: Two vectors are parallel when their cross product is zero. Since their cross product is 4i + 2j – 6k (non-zero), the vectors are not parallel.
To determine if the vectors ((-2, 1, -1)) and ((0, 3, 1)) are parallel, we can check if one vector is a scalar multiple of the other.
Two vectors (mathbf{a} = (a_1, a_2, a_3)) and (mathbf{b} = (b_1, b_2, b_3)) are parallel if there exists a scalar (k) such that:
[
mathbf{a} = k mathbf{b} quad text{or} quad mathbf{b} = k mathbf{a}.
]
For vectors ((-2, 1, -1)) and ((0, 3, 1)):
1. Calculate the ratios:
[
frac{-2}{0}, quad frac{1}{3}, quad frac{-1}{1}.
]
The ratio (frac{-2}{0}) is undefined. Since one of the components of the first vector is undefined when divided by zero, we can conclude that another scalar multiplication to make these two vectors equal is impossible.
Thus, the vectors ((-2, 1, -1)) and ((0, 3, 1)) are not parallel.