Discussion
Which one of the following points lies on the line \(\vec{r} = (\hat{i} + 2\hat{j} - 3\hat{k}) + t(4\hat{i} + 5\hat{j} - 7\hat{k})\) ,\(t \in \mathbb{R}\)?
Answer: D
The point is (9,12,-17).
A general point of the line is \((1 + 4t,\ 2 + 5t,\ -3 - 7t)\). The \(x\)-coordinate 9 needs \(t = 2\), and then:
\[ \begin{aligned} y &= 2 + 5(2) = 12 \\ z &= -3 - 7(2) = -17 \end{aligned} \]
So \((9, 12, -17)\) is on the line. (9,12,-15) has the right \(x\) and \(y\) but the wrong \(z\); (9,15,12) and (12,9,-17) do not come from a single value of \(t\); (-9,-12,17) would need \(t = -2.5\) from \(x\) but \(t = -2.8\) from \(y\).
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