Discussion
Consider the straight line \(\vec{r} = (5\hat{i} + 2\hat{j} - 3\hat{k})\) \(+\, t(4\hat{i} + 6\hat{j} - 7\hat{k}), t \in \mathbb{R}\). Which one of the following points, is a point on the straight line?
Answer: D
The point is (25,32,-38).
A point on the line has the form \((5 + 4t, 2 + 6t, -3 - 7t)\). For the \(x\)-coordinate to be 25, \(t = 5\), and then
\[ (5 + 20,\ 2 + 30,\ -3 - 35) = (25, 32, -38) \]
Testing the others: (21,24,-31) needs \(t = 4\) from \(x\) but gives \(y = 26\); (17,20,-22) needs \(t = 3\) but gives \(z = -24\); (1,-4,5) needs \(t = -1\) but gives \(z = 4\); (45,66,-36) needs \(t = 10\) but gives \(y = 62\).
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