Let \( \mathrm{L_1} \) and \( \mathrm{L_2} \) be any two lines given by
\( \mathrm{L_1}:3\hat{\imath}+2\hat{\jmath}-4\hat{k}+\lambda\left(\hat{\imath}+2\hat{\jmath}+2\hat{k}\right) \)
\( \mathrm{L_2}:5\hat{\imath}-2\hat{\jmath}+\mu\left(3\hat{\imath}+2\hat{\jmath}+6\hat{k}\right) \) Where λ and μ are parameters
then which of the following statements are TRUE?
(A) Equation of a line passing through the point (1, 2, 3) which is parallel to line \( \mathrm{L_1} \) is \( \displaystyle\frac{x-1}{1}=\displaystyle\frac{y-2}{2}=\displaystyle\frac{z-3}{2} \)
(B) Direction cosines of line \( \mathrm{L_2} \) are \( \displaystyle\frac{3}{7},\displaystyle\frac{2}{7},\displaystyle\frac{6}{7} \)
(C) Line \( \mathrm{L_1} \) is parallel to the line \( \mathrm{L_2} \)
(D) The angle between the two lines \( \mathrm{L_1} \) and \( \mathrm{L_2} \) is \( \cos^{-1}\left(\displaystyle\frac{19}{21}\right) \)
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (A), (B) and (D) only
The lines are the vector equations \( \vec{r}=\vec{a_1}+\lambda\vec{b_1} \) and \( \vec{r}=\vec{a_2}+\mu\vec{b_2} \), with direction vectors
\( \displaystyle \vec{b_1}=\hat{\imath}+2\hat{\jmath}+2\hat{k} \)
\( \displaystyle |\vec{b_1}|=\sqrt{1+4+4}=3 \)
\( \displaystyle \vec{b_2}=3\hat{\imath}+2\hat{\jmath}+6\hat{k} \)
\( \displaystyle |\vec{b_2}|=\sqrt{9+4+36}=7 \)
For the angle θ between the lines:
\( \displaystyle \cos\theta=\frac{|\vec{b_1}\cdot\vec{b_2}|}{|\vec{b_1}||\vec{b_2}|} \)
\( \displaystyle =\frac{|3+4+12|}{3\times 7} \)
\( \displaystyle =\frac{19}{21} \)
So the direction ratios are 1, 2, 2 for \( L_1 \) and 3, 2, 6 for \( L_2 \), the direction cosines of \( L_2 \) are \( \frac{3}{7},\frac{2}{7},\frac{6}{7} \), and the angle between the lines is \( \cos^{-1}\left(\frac{19}{21}\right) \).
Checking the statements
(A) Equation of a line passing through the point (1, 2, 3) which is parallel to line \( L_1 \) is \( \frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{2} \) — Correct. It passes through (1, 2, 3) with direction ratios 1, 2, 2, the same as \( L_1 \).
(B) Direction cosines of line \( L_2 \) are \( \frac{3}{7},\frac{2}{7},\frac{6}{7} \) — Correct. The direction ratios 3, 2, 6 divided by \( |\vec{b_2}|=7 \) give these values.
(C) Line \( L_1 \) is parallel to the line \( L_2 \) — Incorrect. The direction ratios 1, 2, 2 and 3, 2, 6 are not proportional (\( \frac{1}{3}\ne\frac{2}{2} \)).
(D) The angle between the two lines \( L_1 \) and \( L_2 \) is \( \cos^{-1}\left(\frac{19}{21}\right) \) — Correct. \( \cos\theta=\frac{19}{3\times 7}=\frac{19}{21} \).