Question

Vector equation of the line \( \frac{x-5}{-4} = \frac{y-3}{5} = \frac{z+3}{-8} \) is:

• A. \( \vec{r} = 4\hat{i} - 5\hat{j} - 8\hat{k} + \mu(5\hat{i} + 3\hat{j} - 3\hat{k}) \)
• B. \( \vec{r} = -4\hat{i} + 5\hat{j} + 8\hat{k} + \mu(5\hat{i} + 3\hat{j} - 3\hat{k}) \)
• C. \( \vec{r} = 5\hat{i} + 3\hat{j} - 3\hat{k} + \mu(4\hat{i} - 5\hat{j} - 8\hat{k}) \)
• D. \( \vec{r} = 5\hat{i} + 3\hat{j} - 3\hat{k} + \mu(-4\hat{i} + 5\hat{j} - 8\hat{k}) \)

Hint:

Watch out for the signs in the point coordinates. \( x-x_1 \) means \( x_1 \) is positive, and \( z+3 \) is \( z-(-3) \), meaning \( z_1 \) is negative.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
To convert a Cartesian equation \( \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} \) to vector form \( \vec{r} = \vec{a} + \mu\vec{b} \), identify:
1. The point \( (x_1, y_1, z_1) \).
2. The direction ratios \( (a, b, c) \).
Step 2: Detailed Explanation:
From the Cartesian equation \( \frac{x-5}{-4} = \frac{y-3}{5} = \frac{z+3}{-8} \):
1. The point on the line is \( (5, 3, -3) \). Its position vector is \( \vec{a} = 5\hat{i} + 3\hat{j} - 3\hat{k} \).
2. The direction ratios are \( (-4, 5, -8) \). The direction vector is \( \vec{b} = -4\hat{i} + 5\hat{j} - 8\hat{k} \).
Substitute these into the vector equation form:
\[ \vec{r} = (5\hat{i} + 3\hat{j} - 3\hat{k}) + \mu(-4\hat{i} + 5\hat{j} - 8\hat{k}) \] Step 3: Final Answer:
The vector equation matches option (D).