Discussion
The vector form of the straight line \(\frac{x - 2}{1} = \frac{y - 1}{-1} = \frac{z - 1}{-2}\) is
Answer: E
The vector form is \(\vec{r} = (2 + \mu)\hat{i} + (1 - \mu)\hat{j} + (1 - 2\mu)\hat{k}\).
Set each ratio equal to \(\mu\): \(x - 2 = \mu\), \(y - 1 = -\mu\), \(z - 1 = -2\mu\). So x = 2 + μ, y = 1 - μ, z = 1 - 2μ: the point (2, 1, 1) plus μ times the direction (1, -1, -2).
Each other option gets one component wrong: \(1 + 2\mu\) and \(1 + \mu\) flip a sign, \(1 - 2\mu\) in the y-component uses the z-ratio, and \(2 + 3\mu\) uses a direction ratio of 3 that is not in the line.
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