The vector equation of the line passing through the point (−2, 4, 1) and parallel to the z-axis is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\vec{r} = (-2\hat{\imath}+4\hat{\jmath}+\hat{k})+\lambda\hat{k}\)
The line through the point with position vector \(\vec{a}\) and parallel to \(\vec{b}\) is \(\vec{r} = \vec{a}+\lambda\vec{b}\).
Here the point gives \(\vec{a} = -2\hat{\imath}+4\hat{\jmath}+\hat{k}\), and a line parallel to the z-axis has direction \(\vec{b} = \hat{k}\):
\(\vec{r} = (-2\hat{\imath}+4\hat{\jmath}+\hat{k})+\lambda\hat{k}\)
In Options (1), (2) and (4) the direction vector is not a multiple of \(\hat{k}\), so none of those lines is parallel to the z-axis.