Question

If \( \vec{A} = 3\hat{i} - 2\hat{j} + \hat{k} \), \( \vec{B} = \hat{i} - 3\hat{j} + 5\hat{k} \) and \( \vec{C} = 2\hat{i} + \hat{j} - 4\hat{k} \) form a right angled triangle, then out of the following which one is satisfied?

• A. \( \vec{B} = \vec{A} + \vec{C}, \; B^2 = A^2 + C^2 \)
• B. \( \vec{A} = \vec{B} + \vec{C}, \; B^2 = A^2 - C^2 \)
• C. \( \vec{C} = \vec{A} + \vec{B}, \; C^2 = A^2 + B^2 \)
• D. \( \vec{A} = \vec{B} + \vec{C}, \; B^2 = A^2 + C^2 \)

Hint:

For vectors forming a right angled triangle, always verify both vector addition and Pythagoras theorem.

Answer 1

The Correct Option is D

Step 1: Use condition for right angled triangle.
If vectors form a right angled triangle, then the square of the hypotenuse equals the sum of squares of the other two sides.

Step 2: Check vector addition.
\[ \vec{B} + \vec{C} = (1+2)\hat{i} + (-3+1)\hat{j} + (5-4)\hat{k} = 3\hat{i} - 2\hat{j} + \hat{k} \] \[ \vec{B} + \vec{C} = \vec{A} \]

Step 3: Verify magnitude relation.
\[ A^2 = B^2 + C^2 \] Thus the given condition is satisfied.