Question

Show that the vectors  2i - j + k, i - 3j - 5k  and  3i - 4j - 4k form the vertices of a right angled triangle.

Answer 1

Let A = 2i - j + k B = i - 3j - 5k and C = 3i - 4j - 4k

vector AB = (i - 3j - 5k) - (2i - j + k) = -i - 2j - 6k

⇒ |vector AB| = √(1 + 4 + 36) = √41

vector BC = (3i - 4j - 4k) - (i - 3j - 5k) = 2i - j + k

⇒ |vector BC| = √(4 + 1 + 1) = √6

and vector AC = (3i - 4j - 4k) - (2i - j + k) = i - 3j - 5k

⇒|vector AC| = √(1 + 9 + 25) = √35

∴ |vector AB|² = |vector AC|² + |vector  BC|²

Hence, ABC is a right angled triangle.