Show that b → × c → · a → × d → + c → × a → · b → × d → + a → × b → · c → × d → = 0

Given, a → , b → , c → and d → are vectors. To prove, b → × c → · a → × d → + c → × a → · b → × d → + a → × b → · c → × d → = 0 Now, take b → × c → · a → × d → = b → c → a → × d → = b → · c → × a → × d → = b → · c → · d → a → - c → · a → d → = b → · a → c → · d → - b → · d → c → · a → … 1 Similarly, c → × a → · b → × d → = b → · c → a → · d → - c → · d → b → · a → … 2 and a → × b → · c → × d → = a → · c → b → · d → - a → · d → b → · c → … 3 add equation 1 , 2 and 3 b → × c → · a → × d → + c → × a → · b → × d → + a → × b → · c → × d → = 0 Hence, proved.

Prove that i ^ × a → × i ^ + j ^ × a → × j ^ + k ^ × a → × k ^ = 2 a → .

Given, a → is any vector. To show that: i ^ × a → × i ^ + j ^ × a → × j ^ + k ^ × a → × k ^ = 2 a → Now, [using vector triple product] , i ^ × a → × i ^ = i ^ · i ^ a → - i ^ · a → i ^ = a → - i ^ · a → i ^ … 1 Similarly, j ^ × a → × j ^ = a → - j ^ · a → j ^ … 2 and k ^ × a → × k ^ = a → - k ^ · a → k ^ … 3 Also, we know, a → = i ^ · a → i ^ + j ^ · a → j ^ + k ^ · a → k ^ Now, add equation 1 , 2 and 3 . i ^ × a → × i ^ + j ^ × a → × j ^ + k ^ × a → × k ^ = 2 a → .

Prove that a → × b → a → × c → d → = a · → d → a → b → c →

Given a → , b → , c → and d → are any vectors. To show that: ⇒ a → × b → a → × c → d → = a · → d → a → b → c → ⇒ a → × b → a → × c → d → = d → · a → × b → × a → × c → ⇒ a → × b → a → × c → d → = d → · a → b → c → a → - a → b → a → c → ⇒ a → × b → a → × c → d → = d → · a → b → c → a → since a → b → a → = 0 ⇒ a → × b → a → × c → d → = a · → d → a → b → c →

If a → , b → and c → are non-coplanar unit vectors such that a → × b → × c → = b → + c → 2 , b → c → are non-parallel, then prove that the angle between a → and b → is 3 π 4 .

Given, a → , b → and c → are non-coplanar unit vectors. Such that ⇒ a → × b → × c → = b → + c → 2 To prove that angle between a → and b → is 3 π 4 . ⇒ a → × b → × c → = b → + c → 2 ⇒ a → · c → b → - a → · b → c → = b → + c → 2 [using vector triple product] ⇒ a → · b → = - 1 2 [comparing both sides] ⇒ | a → | · | b → | cos θ = - 1 2 Since a → , b → and c → are unit vectors , a → = b → = 1 ⇒ cos θ = - 1 2 ⇒ θ = 3 π 4

Find a set of vectors reciprocal to the set of vectors - i ^ + j ^ + k ^ , i ^ - j ^ + k ^ , i ^ + j ^ + k ^ .

Given, a → = - i ^ + j ^ + k ^ b → = i ^ - j ^ + k ^ c → = i ^ + j ^ + k ^ To find, reciprocal of the three vectors a → , b → and c → . ⇒ a → ' = b → × c → a → b → c → ⇒ a → ' = 2 - i ^ + k ^ 4 ⇒ a → ' = 1 2 - i ^ + k ^ ⇒ b → ' = c → × a → a → b → c → ⇒ b → ' = 2 - j ^ + k ^ 4 ⇒ b → ' = 1 2 - j ^ + k ^ ⇒ c → ' = a → × b → a → b → c → ⇒ c → ' = 2 i ^ + j ^ 4 ⇒ c → ' = 1 2 i ^ + j ^ .

Solve r → × b → = a → , where a → and b → are given vectors such that a → · b → = 0 .

Given: a → · b → = 0 , r → × b → = a → ⇒ r → = x a → + y b → + z a → × b → , where a → , b → and a → × b → are non - coplanar vectors and x , y , z are constants Now, ⇒ r → × b → = a → ⇒ x a → + y b → + z a → × b → × b → = a → ⇒ x a → × b → + y b → × b → + z a → × b → × b → = a → ⇒ x a → × b → - a → z b → 2 + 1 = 0 Equating vectors we get, x = 0 , z = - 1 b → 2 . Hence, r → = y b → + - 1 b → 2 a → × b → .

Find vector r → if r → · a → = m and r → × b → = c → , where a → , b → ≠ 0 .

Given: r → · a → = m r → × b → = c → To find vector r → , when a → , b → ≠ 0 ⇒ r → × b → = c → ⇒ r → × b → × a → = c → × a → ⇒ r → · a → · b → - b → · a → · r → = c → × a → ⇒ m b → - b → · a → · r → = c → × a → ⇒ b → · a → r → = m b → - c → × a → ⇒ r → = m b → - c → × a → b → · a →

E M B I B E

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 4

Amit M Agarwal Solutions for Chapter: Product of Vectors, Exercise 4: Exercise for Session 4

Author:Amit M Agarwal

Amit M Agarwal Mathematics Solutions for Exercise - Amit M Agarwal Solutions for Chapter: Product of Vectors, Exercise 4: Exercise for Session 4

Attempt the practice questions on Chapter 2: Product of Vectors, Exercise 4: Exercise for Session 4 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: Product of Vectors, Exercise 4: Exercise for Session 4 with Hints & Solutions

HARD

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 4>Q 3

Show that $(\vec{b} \times \vec{c}) \cdot (\vec{a} \times \vec{d}) + (\vec{c} \times \vec{a}) \cdot (\vec{b} \times \vec{d}) + (\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = 0$

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 4>Q 4

Prove that $\hat{i} \times (\vec{a} \times \hat{i}) + \hat{j} \times (\vec{a} \times \hat{j}) + \hat{k} \times (\vec{a} \times \hat{k}) = 2 \vec{a}$ .

HARD

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 4>Q 5

Prove that
$[\vec{a} \times \vec{b} \vec{a} \times \vec{c} \vec{d}] = (\vec{a \cdot} \vec{d}) [\vec{a} \vec{b} \vec{c}]$

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 4>Q 6

If $\vec{a}, \vec{b}$ and $\vec{c}$ are non-coplanar unit vectors such that $\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\vec{b} + \vec{c}}{\sqrt{2}}$ , $\vec{b}$ $\vec{c}$ are non-parallel, then prove that the angle between $\vec{a}$ and $\vec{b}$ is $\frac{3 π}{4}$ .

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 4>Q 7

Find a set of vectors reciprocal to the set of vectors $- \hat{i} + \hat{j} + \hat{k}, \hat{i} - \hat{j} + \hat{k}, \hat{i} + \hat{j} + \hat{k}$ .

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 4>Q 8

If $\vec{a}, \vec{b}, \vec{c} and {\vec{a}}^{'}, {\vec{b}}^{'}, {\vec{c}}^{'}$ are the reciprocal system of vectors, then prove that ${\vec{a}}^{'} \times {\vec{b}}^{'} + {\vec{b}}^{'} \times {\vec{c}}^{'} + {\vec{c}}^{'} \times {\vec{a}}^{'} = \frac{\vec{a} + \vec{b} + \vec{c}}{[\vec{a} \vec{b} \vec{c}]}$