Discussion
Which one of the following is a point on the straight line \(\vec{r} = (13\hat{i} - 14\hat{j} + 23\hat{k})\) \(+ \lambda(5\hat{i} - 7\hat{j} - 9\hat{k})\) ,\(\lambda \in \mathbb{R}\)
Answer: D
The point is (23,-28,5).
A general point of the line is \((13 + 5\lambda, -14 - 7\lambda, 23 - 9\lambda)\). Taking \(\lambda = 2\):
\[ (13 + 10,\ -14 - 14,\ 23 - 18) = (23, -28, 5) \]
(23, -28, 7) has the right first two coordinates but the wrong third, so it fails the z-check for \(\lambda = 2\). (5, -7, -9) is the direction vector, not a point of the line, and (13, -14, -23) and (13, 14, 23) change the signs of the base point.
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