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:

In vector equations of lines, \( \vec{r_0} \) is the position vector of a point on the line, and \( \vec{v} \) is the direction vector. The parameter \( \mu \) represents the scalar multiple along the line.

Answer 1

The Correct Option is D

Step 1: Parametric Form of the Equation of a Line.
The given equation represents a line in 3D space. We can express the parametric form as: \[ \frac{x - 5}{4} = \frac{y - 3}{5} = \frac{z + 3}{-8} = \mu \] Thus, the parametric equations of the line are: \[ x = 5 + 4\mu, \quad y = 3 + 5\mu, \quad z = -3 - 8\mu \] Step 2: Vector Form.
The vector equation for the line can be written as: \[ \vec{r} = \vec{r_0} + \mu \vec{v} \] Where \( \vec{r_0} = 5\hat{i} + 3\hat{j} - 3\hat{k} \) is the position vector of a point on the line, and \( \vec{v} = 4\hat{i} + 5\hat{j} - 8\hat{k} \) is the direction vector. Step 3: Conclusion.
Thus, the vector equation of the line is: \[ \vec{r} = 5\hat{i} + 3\hat{j} - 3\hat{k} + \mu(-4\hat{i} + 5\hat{j} - 8\hat{k}) \] which corresponds to option (D).