Question

The perimeter of the triangle whose vertices have the position vectors \( \hat{i} + \hat{j} + \hat{k} \), \( 5\hat{i} + 3\hat{j} - 3\hat{k} \), and \( 2\hat{i} + 5\hat{j} + 9\hat{k} \) is

• A. \( \sqrt{15} - \sqrt{157} \) units
• B. \( 15 + \sqrt{157} \) units
• C. \( \sqrt{15} + \sqrt{157} \) units
• D. \( \sqrt{15} + \sqrt{157} \) units

Hint:

Use the distance formula in 3D space to find the length of each side of the triangle, and then sum the lengths to find the perimeter.

Answer 1

The Correct Option is C

Step 1: Use the distance formula to calculate the side lengths.
The distance between two points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) in 3D space is given by: \[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \] We calculate the distances between the three pairs of points.
Step 2: Find the perimeter.
Sum the three distances obtained from the previous step to find the perimeter of the triangle.
Step 3: Conclusion.
The perimeter of the triangle is \( \sqrt{15} + \sqrt{157} \) units.