Find the condition that the three points whose position vectors are a → = a i ^ + b j ^ + c k ^ , b → = i ^ + c j ^ and c → = - i ^ - j ^ are collinear.

Given, a → = a i ^ + b j ^ + c k ^ , b → = i ^ + c j ^ , c → = - i ^ - j ^ To find, The condition for the vectors to be collinear A B → = 1 - a i ^ + c - b j ^ - c k ^ B C → = - 2 i ^ + - 1 - c j ^ A C → = - 1 - a i ^ + - 1 - b j ^ - c k ^ If A , B , C are collinear then A B → = λ B C → ⇒ 1 - a i ^ + c - b j ^ - c k ^ = λ - 2 i ^ + - 1 - c j ^ ⇒ 1 - a = - 2 λ , c - b = λ - 1 - c , - c = 0 c = 0 and - b = λ - 1 ⇒ λ = b ⇒ 1 - a = - 2 λ ⇒ a - 2 b = 1 .

Vectors a → and b → are non-collinear. Find for what values of x , vectors c → = x - 2 a → + b → and d → = 2 x + 1 a → - b → are collinear?

Given: a → and b → are non-collinear c → = x - 2 a → + b → d → = 2 x + 1 a → - b → To find: x such that they are collinear. For collinear, c → + λ d → = 0 x - 2 a → + b → + λ 2 x + 1 a → - b → = 0 ⇒ x - 2 + λ 2 x + 1 a → + 1 - λ b → = 0 Now using a → & b → are non-collinear so their coefficient must be zero, So, x - 2 + λ 2 x + 1 = 0 . . . ( i ) And, 1 - λ = 0 . . . ( ii ) ⇒ λ = 1 Now, put λ = 1 in equation ( i ) , we get x = 1 3 .

Let a , → b → , c → are three vectors of which every pair is non-collinear. If the vectors a → + b → and b → + c → are collinear with c → and a → respectively, then find a → + b → + c → .

It is given that a → + b → is collinear with c → . So, we get, a → + b → = λ c → ⇒ a → + b → + c → = λ c → + c → (adding both sides c → ) ⇒ a → + b → + c → = 1 + λ c → . . . i It is also given that b → + c → is collinear with a → . So, we get, b → + c → = β a → Now, adding both sides a → , we get ⇒ a → + b → + c → = 1 + β a → . . . i i . Equating the above two equations, we have, 1 + λ c → = 1 + β a → Non-collinear vectors are linearly independent. So, 1 + λ = 0 ⇒ λ = - 1 and 1 + β = 0 ⇒ β = - 1 ∴ a → + b → + c → = 1 + λ b → = 1 + β a → ⇒ a → + b → + c → = 1 - 1 a → = 1 - 1 b → = 0 .

Show that the vectors i ^ - j ^ - 2 k ^ , 2 i ^ + 3 j ^ + k ^ & 7 i ^ + 3 j ^ - 4 k ^ are coplanar.

Let a → = i ^ - j ^ - 2 k ^ , b → = 2 i ^ + 3 j ^ + k ^ , c → = 7 i ^ + 3 j ^ - 4 k ^ Vectors are coplanar if a → × b → · c → = 0 a → × b → · c → = 1 - 1 - 2 2 3 1 7 3 - 4 a → × b → · c → = 1 · - 12 - 3 + 1 · - 8 - 7 - 2 6 - 21 ⇒ a → × b → · c → = – 15 – 15 + 30 = 0 .

If the vectors 2 i ^ - j ^ + k ^ , i ^ + 2 j ^ - 3 k ^ & 3 i ^ + a j ^ + 5 k ^ are coplanar, then prove that a = - 4 .

Given, vectors are co-planar. a → = 2 i ^ - j ^ + k ^ , b → = i ^ + 2 j ^ - 3 k ^ , c → = 3 i ^ + a j ^ + 5 k ^ If the above vectors are coplanar, then, a → × b → · c → = 0 . 2 - 1 1 1 2 - 3 3 a 5 = 0 , ⇒ 2 10 + 3 a - - 1 5 + 9 + 1 a - 6 = 0 ⇒ 7 a + 20 + 14 - 6 = 0 ⇒ 7 a = - 28 a = - 4 .

Show that the vectors a → - 2 b → + 3 c → , - 2 a → + 3 b → - 4 c → & - b → + 2 c → are coplanar vector, where a → , b → , c → are non-coplanar vectors.

Given: a → , b → , c → are non co-planar. a → - 2 b → + 3 c → , - 2 a → + 3 b → - 4 c → & - b → + 2 c → To find :vector are co-planar 1 - 2 3 - 2 3 - 4 0 - 1 2 = 1 ( 6 − 4 ) + 2 ( − 4 − 0 ) + 3 ( 2 − 0 ) = 2 − 8 + 6 = 0

If a → , b → and c → are non-coplanar vectors, then prove that the four points 2 a → + 3 b → – c → , a → – 2 b → + 3 c → , 3 a → + 4 b → – 2 c → and a → – 6 b → + 6 c → are coplanar

Given a → , b → , c → are non-coplanar vector. Let O A → = 2 a → + 3 b → - c → , O B → = a → - 2 b → + 3 c → , O C → = 3 a → + 4 b → - 2 c → and O D → = a → - 6 b → + 6 c → then A B → = - a → - 5 b → + 4 c → , B C → = 2 a → + 6 b → - 5 c → and C D → = - 2 a → - 10 b → + 8 c → Vectors are coplanar if A B → × B C → · C D → = 0 - 1 - 5 4 2 6 - 5 - 2 - 10 8 = 0 = ( - 1 ) ( 48 - 50 ) - ( - 5 ) ( 16 - 10 ) + 4 ( - 20 + 12 ) = 2 + 30 - 32 = 0 . Hence, Proved.

E M B I B E

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Vector Algebra Exercise 3

Amit M Agarwal Solutions for Chapter: Vector Algebra, Exercise 6: Vector Algebra Exercise 3

Author:Amit M Agarwal

Amit M Agarwal Mathematics Solutions for Exercise - Amit M Agarwal Solutions for Chapter: Vector Algebra, Exercise 6: Vector Algebra Exercise 3

Attempt the practice questions on Chapter 1: Vector Algebra, Exercise 6: Vector Algebra Exercise 3 with hints and solutions to strengthen your understanding. Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY solutions are prepared by Experienced Embibe Experts.

Questions from Amit M Agarwal Solutions for Chapter: Vector Algebra, Exercise 6: Vector Algebra Exercise 3 with Hints & Solutions

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Exercise for Session 3>Q 4

If three points $A, B$ and $C$ have position vectors $(1, x, 3)$ , $(3, 4, 7)$ and $(y, - 2, - 5)$ , respectively and if they are collinear, then find $(x, y)$ .

EASY

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Exercise for Session 3>Q 5

Find the condition that the three points whose position vectors are $\vec{a} = a \hat{i} + b \hat{j} + c \hat{k}, \vec{b} = \hat{i} + c \hat{j}$ and $\vec{c} = - \hat{i} - \hat{j}$ are collinear.

EASY

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Exercise for Session 3>Q 6

Vectors $\vec{a}$ and $\vec{b}$ are non-collinear. Find for what values of $x$ , vectors $\vec{c} = (x - 2) \vec{a} + \vec{b}$ and $\vec{d} = (2 x + 1) \vec{a} - \vec{b}$ are collinear?

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Exercise for Session 3>Q 7

Let $\vec{a,} \vec{b}, \vec{c}$ are three vectors of which every pair is non-collinear. If the vectors $\vec{a} + \vec{b}$ and $\vec{b} + \vec{c}$ are collinear with $\vec{c}$ and $\vec{a}$ respectively, then find $\vec{a} + \vec{b} + \vec{c} .$

EASY

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Exercise for Session 3>Q 8

Show that the vectors $\hat{i} - \hat{j} - 2 \hat{k}, 2 \hat{i} + 3 \hat{j} + \hat{k} & 7 \hat{i} + 3 \hat{j} - 4 \hat{k}$ are coplanar.

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Exercise for Session 3>Q 9

If the vectors $2 \hat{i} - \hat{j} + \hat{k}, \hat{i} + 2 \hat{j} - 3 \hat{k} & 3 \hat{i} + a \hat{j} + 5 \hat{k}$ are coplanar, then prove that $a = - 4$ .

EASY

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Exercise for Session 3>Q 10

Show that the vectors $(\vec{a} - 2 \vec{b} + 3 \vec{c}), (- 2 \vec{a} + 3 \vec{b} - 4 \vec{c}) & (- \vec{b} + 2 \vec{c})$ are coplanar vector, where $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors.

EASY

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 1 - Vector Algebra>Exercise for Session 3>Q 11

If $\vec{a}, \vec{b}$ and $\vec{c}$ are non-coplanar vectors, then prove that the four points $2 \vec{a} + 3 \vec{b} - \vec{c}$ , $\vec{a} - 2 \vec{b} + 3 \vec{c}$ , $3 \vec{a} + 4 \vec{b} - 2 \vec{c}$ and $\vec{a} - 6 \vec{b} + 6 \vec{c}$ are coplanar

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