Question

Show that the vectors \( 2\hat{i} - \hat{j} + \hat{k}, \hat{i} - 3\hat{j} - 5\hat{k}, 3\hat{i} - 4\hat{j} - 4\hat{k} \) form the vertices of a right-angled triangle. 

Hint:

To prove a right-angled triangle, use the Pythagorean theorem with the magnitudes of the vectors.

Answer 1

Step 1: Find the magnitude of each vector: \[ |v_1| = \sqrt{(2)^2 + (-1)^2 + (1)^2}, \quad |v_2| = \sqrt{(1)^2 + (-3)^2 + (-5)^2}, \quad |v_3| = \sqrt{(3)^2 + (-4)^2 + (-4)^2}. \] 

Step 2: Apply the Pythagorean theorem to check if the sum of squares of two vectors equals the square of the third vector. 

Step 3: If the equation holds, the vectors form a right-angled triangle.