Two points are moving along the lines :
\( L_1:\vec{r}=\left(2\hat{\imath}+3\hat{\jmath}+\hat{k}\right)+\lambda\left(\hat{\imath}+2\hat{\jmath}+\hat{k}\right) \)
\( L_2:\vec{r}=\left(\hat{\imath}+\hat{\jmath}\right)+\mu\left(2\hat{\imath}+\hat{\jmath}-\hat{k}\right) \)
where λ and μ are real parameters , then which of the following statements are correct ?
A. The direction ratios of the line perpendicular to both lines are <−1, 1,−1>.
B. The direction cosines of the second line \( L_2 \) are <\( \frac{2}{\sqrt{6}},\frac{-1}{\sqrt{6}},\frac{-1}{\sqrt{6}} \)>.
C. Angle between two lines is \( \pi/3 \).
D. The lines are skew lines.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → A and C only
From the equations:
\( \displaystyle \vec{a}_1=2\hat{\imath}+3\hat{\jmath}+\hat{k},\quad \vec{b}_1=\hat{\imath}+2\hat{\jmath}+\hat{k} \)
\( \displaystyle \vec{a}_2=\hat{\imath}+\hat{\jmath},\quad \vec{b}_2=2\hat{\imath}+\hat{\jmath}-\hat{k} \)
A line perpendicular to both has the direction of the cross product:
\( \displaystyle \vec{b}_1\times\vec{b}_2=\begin{vmatrix}\hat{\imath} & \hat{\jmath} & \hat{k}\\1 & 2 & 1\\2 & 1 & -1\end{vmatrix} \)
\( \displaystyle =(-2-1)\hat{\imath}-(-1-2)\hat{\jmath}+(1-4)\hat{k} \)
\( \displaystyle =-3\hat{\imath}+3\hat{\jmath}-3\hat{k} \)
The direction cosines of \( L_2 \) divide its direction ratios 2, 1, −1 by \( |\vec{b}_2|=\sqrt{4+1+1}=\sqrt{6} \):
\( \displaystyle \frac{2}{\sqrt{6}},\ \frac{1}{\sqrt{6}},\ \frac{-1}{\sqrt{6}} \)
The angle between the lines:
\( \displaystyle \cos\theta=\frac{|\vec{b}_1\cdot\vec{b}_2|}{|\vec{b}_1|\,|\vec{b}_2|} \)
\( \displaystyle =\frac{|2+2-1|}{\sqrt{6}\,\sqrt{6}} \)
\( \displaystyle =\frac{3}{6}=\frac{1}{2} \)
\( \displaystyle \theta=\frac{\pi}{3} \)
The lines are not parallel. For the shortest distance:
\( \displaystyle \vec{a}_2-\vec{a}_1=-\hat{\imath}-2\hat{\jmath}-\hat{k} \)
\( \displaystyle (\vec{a}_2-\vec{a}_1)\cdot(\vec{b}_1\times\vec{b}_2)=(-1)(-3)+(-2)(3)+(-1)(-3) \)
\( \displaystyle =3-6+3=0 \)
So the common perpendicular has direction ratios −1, 1, −1, the direction cosines of \( L_2 \) are \( \frac{2}{\sqrt{6}},\frac{1}{\sqrt{6}},\frac{-1}{\sqrt{6}} \), the angle is \( \frac{\pi}{3} \), and the shortest distance is 0 (the lines intersect).
Checking the statements
(A) The direction ratios of the line perpendicular to both lines are <−1, 1,−1>. — Correct. \( \vec{b}_1\times\vec{b}_2=-3\hat{\imath}+3\hat{\jmath}-3\hat{k} \), proportional to −1, 1, −1.
(B) The direction cosines of the second line \( L_2 \) are <\( \frac{2}{\sqrt{6}},\frac{-1}{\sqrt{6}},\frac{-1}{\sqrt{6}} \)>. — Incorrect. The second direction cosine is \( +\frac{1}{\sqrt{6}} \), not \( -\frac{1}{\sqrt{6}} \).
(C) Angle between two lines is \( \pi/3 \). — Correct. \( \cos\theta=\frac{3}{6}=\frac{1}{2} \).
(D) The lines are skew lines. — Incorrect. The shortest distance is 0, so the lines intersect.