Prove that the points having position vectors i ^ + 2 j ^ + 3 k ^ , 3 i ^ + 4 j ^ + 7 k ^ , - 3 i ^ - 2 i ^ - 5 k ^ are collinear.

Let A , B , C be the points with position vectors i ^ + 2 j ^ + 3 k ^ , 3 i ^ + 4 j ^ + 7 k ^ , - 3 i ^ - 2 j ^ - 5 k ^ Then, A B → = Position vector of B - Position vector of A = 3 i ^ + 4 j ^ + 7 k ^ - i ^ - 2 j ^ - 3 k ^ = 2 i ^ + 2 j ^ + 4 k ^ B C → = Position vector of C - Position vector of B = - 3 i ^ - 2 j ^ - 5 k ^ - 3 i ^ - 4 j ^ - 7 k ^ = - 6 i ^ - 6 j ^ - 12 k ^ = - 3 2 i ^ + 2 j ^ + 4 k ^ ∴ B C → = - 3 A B → So, A B → and B C → are parallel vectors. But B is a point common to them. So, A B → and B C → are collinear. Hence, A , B , C are collinear.

Prove that the points having position vectors i ^ + 2 j ^ + 3 k ^ , 3 i ^ + 4 j ^ + 7 k ^ , - 3 i ^ - 2 i ^ - 5 k ^ are collinear.

Let A , B , C be the points with position vectors i ^ + 2 j ^ + 3 k ^ , 3 i ^ + 4 j ^ + 7 k ^ , - 3 i ^ - 2 j ^ - 5 k ^ Then, A B → = Position vector of B - Position vector of A = 3 i ^ + 4 j ^ + 7 k ^ - i ^ - 2 j ^ - 3 k ^ = 2 i ^ + 2 j ^ + 4 k ^ B C → = Position vector of C - Position vector of B = - 3 i ^ - 2 j ^ - 5 k ^ - 3 i ^ - 4 j ^ - 7 k ^ = - 6 i ^ - 6 j ^ - 12 k ^ = - 3 2 i ^ + 2 j ^ + 4 k ^ ∴ B C → = - 3 A B → So, A B → and B C → are parallel vectors. But B is a point common to them. So, A B → and B C → are collinear. Hence, A , B , C are collinear.

If a , → b → , c → are two non-collinear vectors prove that the points with position vectors a → + b , → a → - b → and a → + λ b → are collinear for all real values of λ .

Given: a → , b → , c → are non collinear vectors Let the position vectors of points A , B and C be a → + b , → a → - b → , a → + λ b → respectively Then, A B → = Position vector of B - Position vector of A = a → - b → - a → - b → = - 2 b → B C → = Position vector of C - Position vector of B = a → + λ b → - a → + b → = b → λ - 1 C A → = Position vector of A − Position vector of C = a → + b → - a → - λ b → = b → 1 - λ Now, the position vectors are collinear if and only if A B → , B C → , C A → are parallel. So, A B → = β B C → ⇒ - 2 b = β b → λ - 1 ⇒ - 2 = β λ - 1 ⇒ β = - 2 λ - 1 and B C → = - C A → . Hence, for real values of λ , the given position vectors are collinear.

If A O → + O B → = B O → + O C → , prove that A , B , C are collinear points.

We have, A O → + O B → = B O → + O C → ⇒ O B → - O A → = O C → - O B → ⇒ A B → = B C → Hence A , B and C are collinear points.

Show that the vectors 2 i ^ - 3 j ^ + 4 k ^ and - 4 i ^ + 6 j ^ - 8 k ^ are collinear.

Given the position vectors 2 i ^ - 3 j ^ + 4 k ^ and - 4 i ^ + 6 j ^ - 8 k ^ Let a → = 2 i ^ - 3 j ^ + 4 k ^ and b → = - 4 i ^ + 6 j ^ - 8 k ^ Then, b → = - 4 i ^ + 6 j ^ - 8 k ^ = - 2 2 i ^ - 3 j ^ + 4 k ^ = - 2 a → Since a → , b → are parallel, a → , b → are collinear.