Question

Find the distance between the parallel lines given by \( \bar{r}=(2 \hat{i}-\hat{j}+\hat{k})+\lambda(2 \hat{i}+\hat{j}-2 \hat{k}) \) and \( \bar{r}=(\hat{i}-\hat{j}+2 \hat{k})+\mu(2 \hat{i}+\hat{j}-2 \hat{k}) \).

Answer 1

 We are given two parametric equations for the lines:

\(r_1 = (2\hat i - \hat j + \hat k) + \lambda(2\hat i + \hat j - 2\hat k)\)

\(r_2 = (\hat i - \hat j + \hat 2k) + \mu(2\hat i + \hat j - 2\hat k)\)

To find the distance between these two parallel lines, we will use the formula for the distance between two parallel lines:

\(d = \frac{|(r_1 - r_2).(d_1\times d_2)|}{|d_1 \times d_2|}\)

Where:

• r₁​ and r₂​ are points on the two lines.
• d₁​ and d₂​ are the direction vectors of the two lines.
• d₁ × d₂​ is the cross product of the direction vectors, which will be zero if the lines are parallel.

From the equations of the lines:

• The point r₁​ on the first line is \(2\hat i - \hat j + \hat k\)

• The direction vector d₁​ for the first line is \(2\hat i + \hat j - 2 \hat k\)

• The point r₂ on the second line is \(\hat i - \hat j + 2 \hat k\)

• The direction vector d₂​ for the second line is \(2\hat i + \hat j - 2 \hat k.\)

Since both lines share the same direction vector d₁ = d₂ = \(2\hat i + \hat j - 2 \hat k\) the lines are indeed parallel.

Now, we calculate the cross product of \(d_1 = 2\hat i + \hat j - 2 \hat k\) and \(r_1 - r_2 = \hat i - \hat k\)

Using the formula for the distance between two parallel lines:

\(d = \frac{|(r_1 - r_2).(d_1 \times d_2)|}{|d_1 \times d_2|}\)

Since d₁ = d₂​, the magnitude of the cross product d₁ x d₂​ is zero. Therefore, the distance between the lines is:

\(d = \frac{\sqrt2}{3}\)

Thus, the distance between the two parallel lines is \(d = \frac{\sqrt2}{3}\).