Discussion
The straight line \(\vec{r} = (2 - 3t)(\hat{i} + \hat{j})\) + \((6t + 1)(\hat{j} + \hat{k})\) + \((12t - 11)(\hat{k} + \hat{i}), t \in \mathbb{R}\) , is parallel to the vector
Answer: B
The line is parallel to \(3\hat{i} + \hat{j} + 6\hat{k}\).
Collect the terms in \(t\); their coefficient vector is the direction of the line:
\[ \begin{aligned} &-3t(\hat{i} + \hat{j}) + 6t(\hat{j} + \hat{k}) + 12t(\hat{k} + \hat{i}) \\ &= t(9\hat{i} + 3\hat{j} + 18\hat{k}) \\ &= 3t(3\hat{i} + \hat{j} + 6\hat{k}) \end{aligned} \]
Any multiple of \(3\hat{i} + \hat{j} + 6\hat{k}\) is parallel to the line. The other options flip a sign or change a component, so they are not multiples of it.
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