Find the shortest distance between the following lines whose vector equations are :
(i) `vec(r) = ( 8 + 3 lambda) hati - (9 + 16 lambda) hatj + (10 + 7 lambda) hatk `
and `vec(r) = 15 hati + 29 hatj + 5 hatk + mu (3 hati + 8 hatj - 5 hatk)`
(ii) `vec(r) = 3 hati - 15 hatj + 9 hatk + lambda (2 hati - 7 hatj + 5 hatk)`
`and vec(r) = (2mu - 1) hati + (1 + mu ) hatj + (9 - 3mu ) hatk`.

Answer

To find the shortest distance between the given lines, we can use the formula for the shortest distance between two skew lines represented in vector form:

For lines given by:
1. \(\vec{r_1} = \vec{a} + \lambda \vec{b}\)
2. \(\vec{r_2} = \vec{c} + \mu \vec{d}\)

The shortest distance \(D\) between the two lines is given by:
\[ D = \frac{|(\vec{c} - \vec{a}) \cdot (\vec{b} \times \vec{d})|}{|\vec{b} \times \vec{d}|} \]

### Part (i)

**Step 1: Identify the vectors from the equations.**

From the first line:
\[ \vec{r_1} = (8 + 3\lambda) \hat{i} - (9 + 16\lambda) \hat{j} + (10 + 7\lambda) \hat{k} \]
We can identify:
- \(\vec{a} = 8\hat{i} - 9\hat{j} + 10\hat{k}\)
- \(\vec{b} = 3\hat{i} - 16\hat{j} + 7\hat{k}\)

From the second line:
\[ \vec{r_2} = 15\hat{i} + 29\hat{j} + 5\hat{k} + \mu(3\hat{i} + 8\hat{j} - 5\hat{k}) \]
We can identify:
- \(\vec{c} = 15\hat{i} + 29\hat{j} + 5\hat{k}\)
- \(\vec{d} = 3\hat{i} + 8\hat{j} - 5\hat{k}\)

**Step 2: Calculate \(\vec{c} - \vec{a}\).**
\[ \vec{c} - \vec{a} = (15 - 8)\hat{i} + (29 + 9)\hat{j} + (5 - 10)\hat{k} = 7\hat{i} + 38\hat{j} - 5\hat{k} \]

**Step 3: Calculate \(\vec{b} \times \vec{d}\).**
Using the determinant formula:
\[ \vec{b} \times \vec{d} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & -16 & 7 \\ 3 & 8 & -5 \end{vmatrix} \]
Calculating the determinant:
\[ = \hat{i}((-16)(-5) - (7)(8)) - \hat{j}((3)(-5) - (7)(3)) + \hat{k}((3)(8) - (-16)(3)) \]
\[ = \hat{i}(80 - 56) - \hat{j}(-15 - 21) + \hat{k}(24 + 48) \]
\[ = 24\hat{i} + 36\hat{j} + 72\hat{k} \]

**Step 4: Calculate the magnitude of \(\vec{b} \times \vec{d}\).**
\[ |\vec{b} \times \vec{d}| = \sqrt{24^2 + 36^2 + 72^2} = \sqrt{576 + 1296 + 5184} = \sqrt{7056} = 84 \]

**Step 5: Calculate the dot product \((\vec{c} - \vec{a}) \cdot (\vec{b} \times \vec{d})\).**
\[ (\vec{c} - \vec{a}) \cdot (\vec{b} \times \vec{d}) = (7\hat{i} + 38\hat{j} - 5\hat{k}) \cdot (24\hat{i} + 36\hat{j} + 72\hat{k}) \]
\[ = 7 \cdot 24 + 38 \cdot 36 - 5 \cdot 72 = 168 + 1368 - 360 = 1176 \]

**Step 6: Calculate the shortest distance \(D\).**
\[ D = \frac{|1176|}{84} = 14 \]

### Part (ii)

**Step 1: Identify the vectors from the equations.**

From the first line:
\[ \vec{r_1} = (3 + 2\lambda) \hat{i} - (15 - 7\lambda) \hat{j} + (9 + 5\lambda) \hat{k} \]
We can identify:
- \(\vec{a} = 3\hat{i} - 15\hat{j} + 9\hat{k}\)
- \(\vec{b} = 2\hat{i} - 7\hat{j} + 5\hat{k}\)

From the second line:
\[ \vec{r_2} = (2\mu - 1) \hat{i} + (1 + \mu) \hat{j} + (9 - 3\mu) \hat{k} \]
We can identify:
- \(\vec{c} = -\hat{i} + \hat{j} + 9\hat{k}\)
- \(\vec{d} = 2\hat{i} + \hat{j} - 3\hat{k}\)

**Step 2: Calculate \(\vec{c} - \vec{a}\).**
\[ \vec{c} - \vec{a} = (-1 - 3)\hat{i} + (1 + 15)\hat{j} + (9 - 9)\hat{k} = -4\hat{i} + 16\hat{j} \]

**Step 3: Calculate \(\vec{b} \times \vec{d}\).**
Using the determinant formula:
\[ \vec{b} \times \vec{d} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -7 & 5 \\ 2 & 1 & -3 \end{vmatrix} \]
Calculating the determinant:
\[ = \hat{i}((-7)(-3) - (5)(1)) - \hat{j}((2)(-3) - (5)(2)) + \hat{k}((2)(1) - (-7)(2)) \]
\[ = \hat{i}(21 - 5) - \hat{j}(-6 - 10) + \hat{k}(2 + 14) \]
\[ = 16\hat{i} + 16\hat{j} + 16\hat{k} \]

**Step 4: Calculate the magnitude of \(\vec{b} \times \vec{d}\).**
\[ |\vec{b} \times \vec{d}| = \sqrt{16^2 + 16^2 + 16^2} = \sqrt{768} = 16\sqrt{3} \]

**Step 5: Calculate the dot product \((\vec{c} - \vec{a}) \cdot (\vec{b} \times \vec{d})\).**
\[ (\vec{c} - \vec{a}) \cdot (\vec{b} \times \vec{d}) = (-4\hat{i} + 16\hat{j}) \cdot (16\hat{i} + 16\hat{j} + 16\hat{k}) \]
\[ = -4 \cdot 16 + 16 \cdot 16 + 0 = -64 + 256 = 192 \]

**Step 6: Calculate the shortest distance \(D\).**
\[ D = \frac{|192|}{16\sqrt{3}} = \frac{12}{\sqrt{3}} = 4\sqrt{3} \]

### Final Answers
1. The shortest distance for part (i) is **14**.
2. The shortest distance for part (ii) is **\(4\sqrt{3}\)**.