If the volume of a parallelepiped whose adjacent edges are a → = 2 i ^ + 3 j ^ + 4 k ^ , b → = i ^ + α j ^ + 2 k ^ , c → = i ^ + 2 j ^ + α k ^ is 15 , then find the value of α , where α > 0 .

Given a → = 2 i ^ + 3 j ^ + 4 k ^ b → = i ^ + α j ^ + 2 k ^ and c → = i ^ + 2 j ^ + α k ^ Volume of the parallelepiped = [ a → b → c → ] = 15 2 3 4 1 α 2 1 2 α = 15 2 α 2 - 7 α - 9 = 0 2 α - 9 α + 1 = 0 α = 9 2 , α > 0 .

The position vectors of the four angular points of a tetrahedron are A j ^ + 2 k ^ , B 3 i ^ + k ^ , C 4 i ^ + 3 j ^ + 6 k ^ and D 2 i ^ + 3 j ^ + 2 k ^ . Find the volume of the tetrahedron A B C D .

Vertices of tetrahedron are A j ^ + 2 k ^ , B 3 i ^ + k ^ , C 4 i ^ + 3 j ^ + 6 k ^ and D 2 i ^ + 3 j ^ + 2 k ^ Therefore, A B → = Position vector of B → - Position vector of A → ⇒ A B → = 3 i ^ + k ^ - j ^ + 2 k ^ ⇒ A B → = 3 i ^ - j ^ - k ^ Similarly, A C → = 4 i ^ + 2 j ^ + 4 k ^ and A D → = 2 i ^ + 2 j ^ ∴ Volume, V = 1 6 A B → A C → A D → = 1 6 3 - 1 - 1 4 2 4 2 2 0 = 1 6 3 0 - 8 + 0 - 8 - 8 - 4 = 6 cubic unit Since, volume is not negative.

Find the altitude of a parallelepiped whose three coterminous edges are vectors A → = i ^ + j ^ + k ^ , B → = 2 i ^ + 4 j ^ - k ^ and C → = i ^ + j ^ + 3 k ^ , with A → and B → as the sides of the base of the parallelepiped.

Given edges, A → = i ^ + j ^ + k ^ B → = 2 i ^ + 4 j ^ - k ^ C → = i ^ + j ^ + 3 k ^ Volume of parallelepiped = A → B → C → = Base Area × Altitude = 4 Base area = A → × B → = 38 Altitude = A → B → C → A → × B → = 4 38 = 2 38 19 .

Examine whether the vectors a → = 2 i ^ + 3 ȷ ^ + 2 k ^ , b → = i ^ - ȷ ^ + 2 k ^ and c → = 3 i ^ + 2 ȷ ^ + 4 k ^ form a left handed or a right handed system.

Given, a → = 2 i ^ + 3 j ^ + 2 k ^ b → = i ^ - j ^ + 2 k ^ and c → = 3 i ^ + 2 j ^ + 4 k ^ To find whether they are right-handed or left-handed system. a → b → c → = 2 3 2 1 - 1 2 3 2 4 = 2 [ - 4 - 4 ] - 3 [ 4 - 6 ] + 2 [ 2 + 3 ] = - 16 + 6 + 10 = 0 Hence, it is neither a left handed or a right handed system.

Prove that the four points 4 i ^ + 5 j ^ + k ^ , - j ^ - k ^ , 3 i ^ + 9 j ^ + 4 k ^ and 4 - i ^ + j ^ + k ^ are coplanar.

Given, A ≡ 4 i ^ + 5 j ^ + k ^ B ≡ - j ^ - k ^ C ≡ 3 i ^ + 9 j ^ + 4 k ^ D ≡ 4 - i ^ + j ^ + k ^ Then, A B → = Position vector of B → - Position vector of A → ⇒ A B → = - j ^ - k ^ - 4 i ^ + 5 j ^ + k ^ ⇒ A B → = - 4 i ^ - 6 j ^ - 2 k ^ Similarly, A C → = 3 i ^ + 9 j ^ + 4 k ^ - 4 i ^ + 5 j ^ + k ^ ⇒ A C → = - i ^ + 4 j ^ + 3 k ^ And, A D → = 4 - i ^ + j ^ + k ^ - 4 i ^ + 5 j ^ + k ^ ⇒ A D → = - 8 i ^ - j ^ + 3 k ^ If three vectors are coplanar, then scalar triple product of them is zero. Now, A B → A C → A D → - 4 - 6 - 2 - 1 4 3 - 8 - 1 3 = - 4 12 + 3 - - 6 - 3 + 24 - 2 1 + 32 = - 4 15 + 6 21 - 2 33 = - 60 + 126 - 66 = - 126 + 126 = 0 Hence, proved that the given four points are coplanar.

Prove that a → b → c → u → v → w → = a → · u → b → · u → c → · u → a → · v → b → · v → c → · v → a → · w → b → · w → c → · w →

Let, a → = a → 1 i ^ + a → 2 j ^ + a → 2 k ^ b → = b → 1 i ^ + b → 2 j ^ + b → 3 k ^ c → = c → 1 i ^ + c → 2 j ^ + c → 3 k ^ u → = u → 1 i ^ + u → 2 j ^ + u → 3 k ^ v → = v → 1 i ^ + v → 2 j ^ + v → 3 k ^ w → = w → 1 i ^ + w → 2 j ^ + w → 3 k ^ Now, a → b → c → u → v → w → = a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 u 1 u 2 u 3 v 1 v 2 v 3 w 1 w 2 w 2 = a 1 u 1 + a 2 u 2 + a 3 u 3 b 1 u 1 + b 2 u 2 + b 3 u 3 c 1 u 1 + c 2 u 2 + c 3 u 3 a 1 v 1 + a 2 v 2 + a 3 v 3 b 1 v 1 + b 2 v 2 + b 3 v 3 c 1 v 1 + c 2 v 2 + c 3 v 3 a 1 w 1 + a 2 w 2 + a 3 w 3 b 1 w 1 + b 2 w 2 + b 3 w 3 c 1 w 1 + c 2 w 2 + c 3 w 3 = a → · u → b → · u → c → · u → a → · v → b → · v → c → · v → a → · w → b → · w → c → · w → Hence Proved

If a → b → c → = 2 , then find the value of a → + 2 b → - c → a → - b → a → - b → - c →

Given, that a → b → c → = 2 Now, a → + 2 b → - c → a → - b → a → - b → - c → = a → + 2 b → - c → × a → - b → · a → - b → - c → = 2 b → × a → - c → × a → - a → × b → + c → × b → · a → - b → - c → = 3 b → × a → - c → × a → + c → × b → · a → - b → - c → = 3 a → b → c → = 6

If a → , b → and c → are three non-coplanar vectors, then find the value of a → · b → × c → b → · c → × a → + b → · c → × a → c → · a → × b → + c → · a → × b → a → · b → × c →

Given: a → , b → , c → are three non-coplanar vectors. To find: = a → · b → × c → b → · c → × a → + b → · c → × a → c → · a → × b → + c → · a → × b → a → · b → × c → = [ a → b → c → ] [ b → c → a → ] + [ b → c → a → ] [ c → a → b → ] + [ c → a → b → ] [ a → b → c → ] = [ a → b → c → ] [ a → b → c → ] + [ a → b → c → ] [ a → b → c → ] + [ a → b → c → ] [ a → b → c → ] = 3

E M B I B E

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

Amit M Agarwal Solutions for Chapter: Product of Vectors, Exercise 7: Exercise for Session 3

Author:Amit M Agarwal

Amit M Agarwal Mathematics Solutions for Exercise - Amit M Agarwal Solutions for Chapter: Product of Vectors, Exercise 7: Exercise for Session 3

Attempt the practice questions on Chapter 2: Product of Vectors, Exercise 7: Exercise for Session 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: Product of Vectors, Exercise 7: Exercise for Session 3 with Hints & Solutions

MEDIUM

JEE Advanced

IMPORTANT

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

If the volume of a parallelepiped whose adjacent edges are $\vec{a} = 2 \hat{i} + 3 \hat{j} + 4 \hat{k}, \vec{b} = \hat{i} + α \hat{j} + 2 \hat{k}, \vec{c} = \hat{i} + 2 \hat{j} + α \hat{k}$ is $15$ , then find the value of $α$ , where $α > 0$ .

MEDIUM

JEE Advanced

IMPORTANT

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

The position vectors of the four angular points of a tetrahedron are $A (\hat{j} + 2 \hat{k}), B (3 \hat{i} + \hat{k}), C (4 \hat{i} + 3 \hat{j} + 6 \hat{k})$ and $D (2 \hat{i} + 3 \hat{j} + 2 \hat{k})$ . Find the volume of the tetrahedron $A B C D$ .

HARD

JEE Advanced

IMPORTANT

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

Find the altitude of a parallelepiped whose three coterminous edges are vectors $\vec{A} = \hat{i} + \hat{j} + \hat{k}$ , $\vec{B} = 2 \hat{i} + 4 \hat{j} - \hat{k}$ and $\vec{C} = \hat{i} + \hat{j} + 3 \hat{k}$ , with $\vec{A}$ and $\vec{B}$ as the sides of the base of the parallelepiped.

EASY

JEE Advanced

IMPORTANT

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

Examine whether the vectors $\vec{a} = 2 \hat{i} + 3 \hat{ȷ} + 2 \hat{k}, \vec{b} = \hat{i} - \hat{ȷ} + 2 \hat{k}$ and $\vec{c} = 3 \hat{i} + 2 \hat{ȷ} + 4 \hat{k}$ form a left handed or a right handed system.

MEDIUM

JEE Advanced

IMPORTANT

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

Prove that the four points $(4 \hat{i} + 5 \hat{j} + \hat{k}), (- \hat{j} - \hat{k}), (3 \hat{i} + 9 \hat{j} + 4 \hat{k})$ and $4 (- \hat{i} + \hat{j} + \hat{k})$ are coplanar.

MEDIUM

JEE Advanced

IMPORTANT

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

Prove that $[\begin{array}{lll} \vec{a} & \vec{b} & \vec{c} \end{array}] [\begin{array}{l} \vec{u} & \vec{v} & \vec{w} \end{array}] = |\begin{array}{l} \vec{a} \cdot \vec{u} & \vec{b} \cdot \vec{u} & \vec{c} \cdot \vec{u} \\ \vec{a} \cdot \vec{v} & \vec{b} \cdot \vec{v} & \vec{c} \cdot \vec{v} \\ \vec{a} \cdot \vec{w} & \vec{b} \cdot \vec{w} & \vec{c} \cdot \vec{w} \end{array}|$

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 3>Q 9

If $[\vec{a} \vec{b} \vec{c}] = 2$ , then find the value of $[(\vec{a} + 2 \vec{b} - \vec{c}) (\vec{a} - \vec{b}) (\vec{a} - \vec{b} - \vec{c})]$

MEDIUM

JEE Advanced

IMPORTANT

Skills in Mathematics for JEE MAIN & ADVANCED VECTORS & 3D GEOMETRY>Chapter 2 - Product of Vectors>Exercise for Session 3>Q 10

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three non-coplanar vectors, then find the value of $\frac{\vec{a} \cdot (\vec{b} \times \vec{c})}{\vec{b} \cdot (\vec{c} \times \vec{a})} + \frac{\vec{b} \cdot (\vec{c} \times \vec{a})}{\vec{c} \cdot (\vec{a} \times \vec{b})} + \frac{\vec{c} \cdot (\vec{a} \times \vec{b})}{\vec{a} \cdot (\vec{b} \times \vec{c})}$

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