The shortest distance between the lines whose vector equations are r = (6i+2j+2k) + (i-2j+2k) and r = (-4i-k) + (3i-2j-2k) Option 1: 109/12 units Option 2: 12 units Option 3: 25/3 units. Option 4: 9 units

Answer: 9 units. Solution: Here b_1xb_2 = 8i+8j+4k with magnitude 12, and a_2-a_1 = -10i-2j-3k. The dot product is -108, so the shortest distance is, -108, /12 = 9 units.

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The shortest distance between the lines whose vector equations are $\vec{r} = (6\hat{i}+2\hat{j}+2\hat{k}) + \lambda(\hat{i}-2\hat{j}+2\hat{k})$ and $\vec{r} = (-4\hat{i}-\hat{k}) + \mu(3\hat{i}-2\hat{j}-2\hat{k})$

$\frac{109}{12}$ units

12 units

$\frac{25}{3}$ units.

9 units

✅ Correct Option: 4

Here $\vec{b_1}\times\vec{b_2} = 8\hat{i}+8\hat{j}+4\hat{k}$ with magnitude 12, and $\vec{a_2}-\vec{a_1} = -10\hat{i}-2\hat{j}-3\hat{k}$ . The dot product is $-108$ , so the shortest distance is $|-108|/12 = 9$ units.

2025: 22 May Shift 2

Match List-I with List-II

List-I	List-II
Equation of line	Information
(A) $\vec{r} = (3\hat{i} - 2\hat{j} + \hat{k}) + \lambda(\hat{j} - 2\hat{k})$	(I) Direction ratios are 2, 4, -1
(B) $\frac{2-x}{1} = \frac{2y+1}{4}$ , $z = 2$	(II) Perpendicular to $2\hat{i} - \hat{j} + \hat{k}$
(C) $\frac{x}{1} = \frac{y-3}{2} = \frac{3-4z}{2}$	(III) Passing through the point $(3, -2, 1)$
(D) $\vec{r} = (3\hat{i} + 2\hat{j} + \hat{k}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})$	(IV) Direction ratios are -1, 2, 0

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2026: 31 May Shift 2

If the lines $x = ay+b$ , $z = cy+d$ and $x = py+q$ , $z = ry+s$ are perpendicular, then $ap + cr$ is equal to

2022: 6 Aug Shift 2

If two lines $\frac{x-3}{2} = \frac{y-4}{5} = \frac{z}{4}$ and $\frac{x-4}{3} = \frac{y-5}{6} = \frac{1-z}{k}$ , are coplanar, then $k$ is equal to

2025: 13 May Shift 1

If a line makes angles $\alpha, \beta, \gamma$ with the positive directions of $x$ - axis, y -axis, z -axis respectively, then the value of $cos2\alpha + cos2\beta + cos2\gamma$ is equal to

2022: 16 July Shift 2

The two lines given by $\vec{r} = (\hat{i} + 2\hat{j} + 3\hat{k}) + \mu(\hat{i} - \hat{j} + \hat{k})$ and $\vec{r} = (2\hat{i} - \hat{j} - \hat{k}) + \mu(-\hat{i} + \hat{j} - \hat{k})$

A. are perpendicular

B. are parallel.

C. have shortest distance 0.

D. have shortest distance $\sqrt{26}$ .

E. have shortest distance $\sqrt{78}$ .

Which of the above statements are true?

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2026: 29 May Shift 2

If $\lambda$ and $\mu$ are parameters, then

Match the LIST-I with LIST-II

LIST-I		LIST-II
A.	$\vec{r}=2\hat{i}+\hat{j}-3\hat{k}+\lambda(\hat{i}-\hat{j}+2\hat{k})$ parallel to the line	I. $\vec{r}=(\hat{j}+\hat{k})+\mu(-2\hat{i}+\hat{j}-2\hat{k})$
B.	$\vec{r}=\hat{i}+\hat{j}+\lambda(\hat{i}+\hat{j}-2\hat{k})$ perpendicular to the line	II. $\vec{r}=(2\hat{i}+\hat{k})+\mu(-2\hat{i}+\hat{j})$
C.	$\frac{x}{-2}=\frac{y-1}{1}=\frac{z-1}{-2}$ in vector form	III. $\vec{r}=(\hat{i}-\hat{j}+2\hat{k})+\mu(2\hat{i}-2\hat{j}+4\hat{k})$
D.	$\frac{2-x}{2}=y, z=1$ in vector form	IV. $\frac{x-1}{2}=\frac{y+1}{4}=\frac{z}{3}$

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