The shortest distance between the lines `(x-2)/(3)=(y+3)/(4)=(z-1)/(5) and (x-5)/(1)=(y-1)/(2)=(z-6)/(3)`, is

Options

A. 3
B. 2
C. 1
D. 0

Explanation

To find the shortest distance between the two lines given by the equations:

1. \(\frac{x-2}{3} = \frac{y+3}{4} = \frac{z-1}{5}\)
2. \(\frac{x-5}{1} = \frac{y-1}{2} = \frac{z-6}{3}\)

we can express these lines in vector form.

### Step 1: Write the lines in vector form

The vector form of a line can be expressed as:

\[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b} \]

where \(\mathbf{a}\) is a point on the line and \(\mathbf{b}\) is the direction vector.

For the first line:
- Point \(\mathbf{a_1} = (2, -3, 1)\)
- Direction vector \(\mathbf{b_1} = (3, 4, 5)\)

So, the vector equation of the first line is:

\[ \mathbf{r_1} = (2, -3, 1) + \lambda (3, 4, 5) \]

For the second line:
- Point \(\mathbf{a_2} = (5, 1, 6)\)
- Direction vector \(\mathbf{b_2} = (1, 2, 3)\)

So, the vector equation of the second line is:

\[ \mathbf{r_2} = (5, 1, 6) + \mu (1, 2, 3) \]

### Step 2: Use the formula for the shortest distance between two skew lines

The formula for the shortest distance \(d\) between two skew lines is given by:

\[ d = \frac{|\mathbf{a_2} - \mathbf{a_1} \cdot (\mathbf{b_1} \times \mathbf{b_2})|}{|\mathbf{b_1} \times \mathbf{b_2}|} \]

### Step 3: Calculate \(\mathbf{a_2} - \mathbf{a_1}\)

Calculate the vector \(\mathbf{a_2} - \mathbf{a_1}\):

\[ \mathbf{a_2} - \mathbf{a_1} = (5 - 2, 1 + 3, 6 - 1) = (3, 4, 5) \]

### Step 4: Calculate \(\mathbf{b_1} \times \mathbf{b_2}\)

Now, calculate the cross product \(\mathbf{b_1} \times \mathbf{b_2}\):

\[ \mathbf{b_1} = (3, 4, 5), \quad \mathbf{b_2} = (1, 2, 3) \]

Using the determinant for the cross product:

\[ \mathbf{b_1} \times \mathbf{b_2} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 3 & 4 & 5 \\ 1 & 2 & 3 \end{vmatrix} \]

Calculating the determinant:

\[ = \mathbf{i}(4 \cdot 3 - 5 \cdot 2) - \mathbf{j}(3 \cdot 3 - 5 \cdot 1) + \mathbf{k}(3 \cdot 2 - 4 \cdot 1) \]
\[ = \mathbf{i}(12 - 10) - \mathbf{j}(9 - 5) + \mathbf{k}(6 - 4) \]
\[ = 2\mathbf{i} - 4\mathbf{j} + 2\mathbf{k} = (2, -4, 2) \]

### Step 5: Calculate the magnitude of \(\mathbf{b_1} \times \mathbf{b_2}\)

Now, calculate the magnitude:

\[ |\mathbf{b_1} \times \mathbf{b_2}| = \sqrt{2^2 + (-4)^2 + 2^2} = \sqrt{4 + 16 + 4} = \sqrt{24} = 2\sqrt{6} \]

### Step 6: Calculate the dot product \((\mathbf{a_2} - \mathbf{a_1}) \cdot (\mathbf{b_1} \times \mathbf{b_2})\)

Now, calculate the dot product:

\[ (3, 4, 5) \cdot (2, -4, 2) = 3 \cdot 2 + 4 \cdot (-4) + 5 \cdot 2 = 6 - 16 + 10 = 0 \]

### Step 7: Calculate the shortest distance \(d\)

Substituting into the distance formula:

\[ d = \frac{|0|}{2\sqrt{6}} = 0 \]

Thus, the shortest distance between the two lines is \(0\).