Question

The vector equation of the symmetrical form of equation of straight line \( \mathbf{r} = (3\hat{i} + 7\hat{j} + 2\hat{k}) + \lambda (5\hat{i} + 4\hat{j} - 6\hat{k}) \) is:

• A. \( \mathbf{r} = (5\hat{i} + 4\hat{j} - 6\hat{k}) + \mu (3\hat{i} + 7\hat{j} + 2\hat{k}) \)
• B. \( \mathbf{r} = (5\hat{i} - 4\hat{j} + 6\hat{k}) + \mu (3\hat{i} + 7\hat{j} + 2\hat{k}) \)
• C. \( \mathbf{r} = (3\hat{i} + 4\hat{j} - 6\hat{k}) + \mu (5\hat{i} + 7\hat{j} + 2\hat{k}) \)
• D. \( \mathbf{r} = (3\hat{i} + 4\hat{j} + 6\hat{k}) + \mu (5\hat{i} - 7\hat{j} + 2\hat{k}) \)

Hint:

In vector equations, the first term represents a point on the line and the second term represents the direction vector scaled by a parameter.

Answer 1

The Correct Option is A

Step 1: The vector equation of a line is given as: \[ \mathbf{r} = \mathbf{a} + \lambda \mathbf{b}, \] where \( \mathbf{a} \) is a point on the line, and \( \mathbf{b} \) is the direction vector.
Step 2: Comparing the given equation with the standard form, the vector equation in the symmetrical form is: \[ \mathbf{r} = (3\hat{i} + 7\hat{j} + 2\hat{k}) + \mu (5\hat{i} + 4\hat{j} - 6\hat{k}). \]
Final Answer: \[ \boxed{(5\hat{i} + 4\hat{j} - 6\hat{k}) + \mu (3\hat{i} + 7\hat{j} + 2\hat{k})} \]