Discussion
Which one of the following is a vector parallel to the straight line \(\vec{r} = (\hat{i} - 11\hat{j} + 101\hat{k})\) \(+ \lambda(3\hat{i} - 5\hat{j} + 2\hat{k}), \lambda \in \mathbb{R}\)
Answer: A
A parallel vector is \(-3\hat{i} + 5\hat{j} - 2\hat{k}\).
In \(\vec{r} = \vec{a} + \lambda\vec{b}\), the line is parallel to \(\vec{b} = 3\hat{i} - 5\hat{j} + 2\hat{k}\), and any non-zero multiple of \(\vec{b}\) is parallel too. Option A is \(-\vec{b}\).
\(3\hat{i} + 5\hat{j} + 2\hat{k}\) changes only one sign, so it is not a multiple. \(\hat{i} - 11\hat{j} + 101\hat{k}\) is the position vector of a point on the line, not its direction, and the other two are not multiples of \(\vec{b}\).
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