Question

A unit vector perpendicular to the plane formed by the points \( (1, 0, 1) \), \( (0, 2, 2) \), and \( (3, 3, 0) \) is:

• A. \( \frac{1}{\sqrt{5}} (5\hat{i} - \hat{j} - 7\hat{k}) \)
• B. \( \frac{1}{\sqrt{3}} (5\hat{i} + 7\hat{k}) \)
• C. \( \frac{1}{\sqrt{3}} (5\hat{i} + 7\hat{j} + 7\hat{k}) \)
• D. None of these

Hint:

To find a unit vector perpendicular to a plane, compute the cross product of two vectors on the plane and normalize the result.

Answer 1

The Correct Option is A

Step 1: The unit vector perpendicular to a plane is given by the cross product of two vectors lying on the plane.
Step 2: Using the given points, we form two vectors and compute their cross product. After normalizing the result, we get \( \frac{1}{\sqrt{5}} (5\hat{i} - \hat{j} - 7\hat{k}) \).

Final Answer: \[ \boxed{\frac{1}{\sqrt{5}} (5\hat{i} - \hat{j} - 7\hat{k})} \]