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.

E M B I B E

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE

R D Sharma Solutions for Chapter: Algebra of Vectors, Exercise 7: EXERCISE

Author:R D Sharma

R D Sharma Mathematics Solutions for Exercise - R D Sharma Solutions for Chapter: Algebra of Vectors, Exercise 7: EXERCISE

Attempt the free practice questions on Chapter 4: Algebra of Vectors, Exercise 7: EXERCISE with hints and solutions to strengthen your understanding. MATHEMATICS CLASS XII VOLUME-2 solutions are prepared by Experienced Embibe Experts.

Questions from R D Sharma Solutions for Chapter: Algebra of Vectors, Exercise 7: EXERCISE with Hints & Solutions

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 4

If the points with position vectors $10 \hat{i} + 3 \hat{j}, 12 \hat{i} - 5 \hat{j}$ and $a \hat{i} + 11 \hat{j}$ are collinear, find the value of $a$ .

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 4 - Algebra of Vectors>EXERCISE>Q 5

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