Show that the points A, B and C with position vectors, a = 3i - 4j -

1.	Show that the points A, B and C with position vectors, a = 3i - 4j - 4k, vector = 2i - j + k and vector c = i - 3j - 5k respectively form the vertices of a right angled triangle.
Answer» Position vectors of points A, B and C are respectively given as vector a = 3i - 4j - 4k, vector b = 2i - j + k and vector c = i - 3j - 5k Now, vector (AB) = vector b - vector a = 2i - j + k - 3i + 4j = - i + 3j + 5k ⇒ \|vector AB\|² = 1 + 9 + 25 = 35 vector BC = vector c - vector b = i - 3j - 5k - 2i + j - k = - i - 2j - 6k ⇒ \|vector BC\|² = 1 + 4 + 36 = 41 vector CA = vector a - vector c = 3i - 4j - 4k - i + 3j + 5k = 2i - j + k ⇒ \|vector CA\|² = 4 + 1 + 1 = 6 ⇒ \|vector BC\|² = \|vector AB\|² + \|vector CA\|² Hence it form the vertices of a right angled triangle.

Show that the points A, B and C with position vectors, a = 3i - 4j - 4k, vector = 2i - j + k and vector c = i - 3j - 5k respectively form the vertices of a right angled triangle.

Answer»

Position vectors of points A, B and C are respectively given as

vector a = 3i - 4j - 4k, vector b = 2i - j + k and vector c = i - 3j - 5k

Now, vector (AB) = vector b - vector a = 2i - j + k - 3i + 4j = - i + 3j + 5k

⇒ |vector AB|² = 1 + 9 + 25 = 35

vector BC = vector c - vector b = i - 3j - 5k - 2i + j - k = - i - 2j - 6k

⇒ |vector BC|² = 1 + 4 + 36 = 41

vector CA = vector a - vector c = 3i - 4j - 4k - i + 3j + 5k = 2i - j + k

⇒ |vector CA|² = 4 + 1 + 1 = 6

⇒ |vector BC|² = |vector AB|² + |vector CA|²

Hence it form the vertices of a right angled triangle.

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