The equation of line passing through origin and parallel to the line \(\vec{r} = 3\hat{\imath} + 4\hat{\jmath} - 5\hat{k} + t\left(2\hat{\imath} - \hat{\jmath} + 7\hat{k}\right)\), where t is a parameter, is:
(A) \(\displaystyle\frac{x}{2} = \frac{y}{-1} = \frac{z}{7}\)
(B) \(\vec{r} = m\left(12\hat{\imath} - 6\hat{\jmath} + 42\hat{k}\right)\); where m is the parameter
(C) \(\vec{r} = \left(12\hat{\imath} - 6\hat{\jmath} + 42\hat{k}\right) + s\left(0\hat{\imath} - 0\hat{\jmath} + 0\hat{k}\right)\); where s is the parameter
(D) \(\displaystyle\frac{x-3}{0} = \frac{y-4}{0} = \frac{z+5}{0}\)
(E) \(\displaystyle\frac{x}{3} = \frac{y}{4} = \frac{z}{5}\)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
The given line has direction vector
\(\vec{b} = 2\hat{\imath} - \hat{\jmath} + 7\hat{k}\)
Parallel lines have the same direction, and any non-zero multiple of \(\vec{b}\) is also a direction vector. The line through the origin parallel to it is
\(\vec{r} = \lambda\left(2\hat{\imath} - \hat{\jmath} + 7\hat{k}\right)\)
or, in Cartesian form,
\( \displaystyle \frac{x}{2} = \frac{y}{-1} = \frac{z}{7} \)
So the required line passes through (0, 0, 0) and its direction ratios are proportional to 2, −1, 7.
Checking the statements
(A) \(\frac{x}{2} = \frac{y}{-1} = \frac{z}{7}\) — Correct. Through the origin, with direction ratios 2, −1, 7.
(B) \(\vec{r} = m\left(12\hat{\imath} - 6\hat{\jmath} + 42\hat{k}\right)\) — Correct. m = 0 gives the origin, and \(12\hat{\imath} - 6\hat{\jmath} + 42\hat{k} = 6\vec{b}\).
(C) \(\vec{r} = \left(12\hat{\imath} - 6\hat{\jmath} + 42\hat{k}\right) + s\left(0\hat{\imath} - 0\hat{\jmath} + 0\hat{k}\right)\) — Incorrect. Its direction vector is the zero vector, so it gives only the single point (12, −6, 42), not a line.
(D) \(\frac{x-3}{0} = \frac{y-4}{0} = \frac{z+5}{0}\) — Incorrect. It passes through (3, 4, −5), not the origin, and all its direction ratios are 0.
(E) \(\frac{x}{3} = \frac{y}{4} = \frac{z}{5}\) — Incorrect. Direction ratios 3, 4, 5 are not proportional to 2, −1, 7.