Question

The shortest distance between the straight line \[ \vec{r} = 3\hat{i}+12\hat{j}+5\hat{k} +\lambda(2\hat{i}), \qquad \lambda\in\mathbb{R}, \] and the \(x\)-axis is:

• A. 7
• B. 12
• C. \(3\sqrt{5}\)
• D. \(5\sqrt{3}\)
• E. 13

Hint:

When a line's direction vector is \((1, 0, 0)\), the line is horizontal (parallel to x-axis). The shortest distance to the x-axis is simply the Euclidean distance of its y and z coordinates from the origin in the YZ-plane.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The equation \(\vec{r} = (3, 12, 5) + \lambda(2, 0, 0)\) represents a line passing through \((3, 12, 5)\) parallel to the x-axis (since the direction vector is \(\hat{i}\)). The x-axis itself is a line passing through the origin \((0, 0, 0)\) with direction vector \(\hat{i}\). Since both lines are parallel, the distance between them is constant.

Step 2: Key Formula or Approach:
The shortest distance between two parallel lines is the distance from any point on one line to the other line.

Step 3: Detailed Explanation:
1. Let point \(P = (3, 12, 5)\) on the given line.
2. The distance from point \((x, y, z)\) to the x-axis is given by \(\sqrt{y^2 + z^2}\).
3. For point \(P(3, 12, 5)\): \[ \text{Distance} = \sqrt{12^2 + 5^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]

Step 4: Final Answer:
The shortest distance is 13 units.