Question

What is the distance between \[\vec r=(\hat i+\hat j)+\lambda(2\hat i-\hat j+\hat k)\] and \[\vec r=(2\hat i+\hat j-\hat k)+\mu(3\hat i-5\hat j+2\hat k)\]?

Options

• \[\frac{10}{\sqrt{59}}\]

• \[\frac{7}{\sqrt{59}}\]

• \[\frac{10}{59}\]

• \[\frac{\sqrt{59}}{10}\]

Answer 1

For the two lines, \[ \vec{a}_{2} - \vec{a}_{1} = (2\hat{i} + \hat{j} - \hat{k}) - (\hat{i} + \hat{j}) = \hat{i} - \hat{k} \] and \[ \vec{b}_{1} \times \vec{b}_{2} = (2\hat{i} - \hat{j} + \hat{k}) \times (3\hat{i} - 5\hat{j} + 2\hat{k}) = 3\hat{i} - \hat{j} - 7\hat{k}. \]

Using the shortest distance formula,

\[ d = \frac{|(\vec{a}_{2} - \vec{a}_{1}) \cdot (\vec{b}_{1} \times \vec{b}_{2})|}{|\vec{b}_{1} \times \vec{b}_{2}|} \] \[ = \frac{|(\hat{i} - \hat{k}) \cdot (3\hat{i} - \hat{j} - 7\hat{k})|}{\sqrt{3^{2} + (-1)^{2} + (-7)^{2}}} = \frac{|3 + 7|}{\sqrt{59}} \] \[ d = \frac{10}{\sqrt{59}} \]