Question

The shortest distance between the lines, where the first line passes through \((0,0,0)\) and \((2,0,3)\) and the second line passes through \((2,5,0)\) and \((0,4,0)\) is

• A. \(\frac{24}{7}\) units
• B. \(\frac{9}{7}\) units
• C. \(\frac{1}{7}\) units
• D. \(\frac{36}{7}\) units

Hint:

Find direction vectors of both lines, take their cross product, and project the vector joining points on the lines.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The shortest distance between two skew lines is \(d = \dfrac{|(\bar{a}_2 - \bar{a}_1)\cdot(\bar{d}_1\times\bar{d}_2)|}{|\bar{d}_1\times\bar{d}_2|}\).

Step 2: Find the directions.
Line 1 through \((0,0,0)\) and \((2,0,3)\): \(\bar{d}_1 = (2, 0, 3)\). Line 2 through \((2,5,0)\) and \((0,4,0)\): \(\bar{d}_2 = (-2, -1, 0)\).

Step 3: Cross product.
\[ \bar{d}_1\times\bar{d}_2 = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 0 & 3 \\ -2 & -1 & 0 \end{vmatrix} = (0 + 3)\hat{i} - (0 + 6)\hat{j} + (-2 - 0)\hat{k} = (3, -6, -2) \]
Its magnitude is \(\sqrt{9 + 36 + 4} = 7\).

Step 4: Joining vector.
\(\bar{a}_2 - \bar{a}_1 = (2, 5, 0)\). The dot product with the cross product is \(6 - 30 + 0 = -24\).

Step 5: Distance.
\[ d = \frac{|-24|}{7} = \frac{24}{7}\text{ units} \]

Final Answer:
The shortest distance is \(\dfrac{24}{7}\) units, option (A). \[ \boxed{\frac{24}{7}} \]