Question

Find the shortest distance between the lines \[ \mathbf{r} = \hat{i} + \hat{j} + \lambda (2\hat{i} - \hat{j} + \hat{k}) \] \[ \mathbf{r} = 2\hat{i} + \hat{j} - \hat{k} + \mu (3\hat{i} - 5\hat{j} + 2\hat{k}) \]

Hint:

To find the shortest distance between skew lines, use the formula \[ d = \frac{|(\mathbf{PQ} \cdot (\mathbf{d_1} \times \mathbf{d_2}))|}{|\mathbf{d_1} \times \mathbf{d_2}|} \] where \( \mathbf{PQ} \) is a vector connecting points on each line, and \( \mathbf{d_1}, \mathbf{d_2} \) are direction vectors.

Answer 1

Step 1: Identify direction vectors and points. For line 1: \begin{itemize} \item Point \( P(1,1,0) \) \item Direction vector \( \mathbf{d_1} = (2,-1,1) \) \end{itemize} For line 2: \begin{itemize} \item Point \( Q(2,1,-1) \) \item Direction vector \( \mathbf{d_2} = (3,-5,2) \) \end{itemize} Step 2: Compute vector \( \mathbf{PQ} \). \[ \mathbf{PQ} = (2-1, 1-1, -1-0) = (1,0,-1) \] Step 3: Compute shortest distance formula. The formula for shortest distance between two skew lines is: \[ d = \frac{|(\mathbf{PQ} \cdot (\mathbf{d_1} \times \mathbf{d_2}))|}{|\mathbf{d_1} \times \mathbf{d_2}|} \] First, compute \( \mathbf{d_1} \times \mathbf{d_2} \): \[ \mathbf{d_1} \times \mathbf{d_2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k}
2 & -1 & 1
3 & -5 & 2 \end{vmatrix} \] \[ = \hat{i}((-1)(2) - (1)(-5)) - \hat{j}((2)(2) - (1)(3)) + \hat{k}((2)(-5) - (-1)(3)) \] \[ = \hat{i}( -2 + 5 ) - \hat{j}(4 - 3) + \hat{k}(-10 + 3) \] \[ = \hat{i}(3) - \hat{j}(1) + \hat{k}(-7) \] \[ = (3, -1, -7) \] Step 4: Compute \( \mathbf{PQ} \cdot (\mathbf{d_1} \times \mathbf{d_2}) \). \[ (1,0,-1) \cdot (3,-1,-7) = (1)(3) + (0)(-1) + (-1)(-7) \] \[ = 3 + 0 + 7 = 10 \] Step 5: Compute magnitude of \( \mathbf{d_1} \times \mathbf{d_2} \). \[ |\mathbf{d_1} \times \mathbf{d_2}| = \sqrt{3^2 + (-1)^2 + (-7)^2} \] \[ = \sqrt{9 + 1 + 49} = \sqrt{59} \] Step 6: Compute shortest distance. \[ d = \frac{|10|}{\sqrt{59}} \] \[ d = \frac{10}{\sqrt{59}} \] \[ d = \frac{10\sqrt{59}}{59} \] Final Answer: \[ d = \frac{10\sqrt{59}}{59} \]