Match List I with the List II and select the correct answer using the code given below the lists - List I List II- A Volume of parallelepiped determined by vectors a-b and c is 2-Then the volume P 100- of the parallelepiped determined by vectors 2a-b-3b-c and c-a is- B Volume of parallelepiped determined by vectors a-b and c is 5-Then the volume Q 30- of the parallelepiped determined by vectors 3a- b-b- c and 2c - a is- C Area of a triangle with adjacent sides determined by vectors a-b is 20- Then the R 24- area of the triangle with adjacent sides determined by vectors 2a- 3b and a-b is- D Area of a parallelogram with adjacent sides determined by vectors a and b is 30- S 60- Then the area of the parallelogram with adjacent sides determined by vectors- a- b and a is- -Which of the following is the only CORRECT combination-

(A) [a b c]=2 ∴[2(a×b) 3(b×c) (c×a)]=2(a×b) ·[3(b×c) ×(c×a)] =6(a×b) ·[d×(c×a)](let d=b×c) =6(a×b) ·[(d·a)c (d·c)a] =6[a b c](d ·a) 6[a b 0](d·c) =6[a b c][b c ...

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Byju's Answer

Standard XII

Physics

Vectors and Its Types

Match List I ...

Question

Match $List I$ with the $List II$ and select the correct answer using the code given below the lists :

$\begin{matrix} List I & List II (A) & Volume of parallelepiped determined by vectors \to a, \to b and \to c is 2. Then the volume & (P) & 100 of the parallelepiped determined by vectors 2 (\to a \times \to b), 3 (\to b \times \to c) and (\to c \times \to a) is (B) & Volume of parallelepiped determined by vectors \to a, \to b and \to c is 5. Then the volume & (Q) & 30 of the parallelepiped determined by vectors 3 (\to a + \to b), (\to b + \to c) and 2 (\to c + \to a) is (C) & Area of a triangle with adjacent sides determined by vectors \to a, \to b is 20. Then the & (R) & 24 area of the triangle with adjacent sides determined by vectors (2 \to a + 3 \to b) and (\to a - \to b) is (D) & Area of a parallelogram with adjacent sides determined by vectors \to a and \to b is 30. & (S) & 60 Then the area of the parallelogram with adjacent sides determined by vectors (\to a + \to b) and \to a is \end{matrix}$

Which of the following is the only CORRECT combination?

(A) \to (S), (B) \to (Q), (C) \to (R), (B) \to (P)

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(A) \to (Q), (B) \to (R), (C) \to (P), (B) \to (S)

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(A) \to (R), (B) \to (S), (C) \to (P), (B) \to (Q)

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(A) \to (P), (B) \to (S), (C) \to (R), (B) \to (Q)

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Solution

The correct option is C $(A) \to (R), (B) \to (S), (C) \to (P), (B) \to (Q)$
$(A)$
$[\to a \to b \to c] = 2$
$∴ [2 (\to a \times \to b) 3 (\to b \times \to c) (\to c \times \to a)] = 2 (\to a \times \to b) \cdot [3 (\to b \times \to c) \times (\to c \times \to a)]$
$= 6 (\to a \times \to b) \cdot [\to d \times (\to c \times \to a)]$ (let $\to d = \to b \times \to c$ )
$= 6 (\to a \times \to b) \cdot [(\to d \cdot \to a) \to c - (\to d \cdot \to c) \to a]$
$= 6 [\to a \to b \to c] (d \cdot \to a) - 6 [\to a \to b 0] (\to d \cdot \to c)$
$= 6 [\to a \to b \to c] [\to b \to c \to a]$
$= 6 {[\to a \to b \to c]}^{2} = 24$

$(B)$
$[\to a \to b \to c] = 5$
$∴ [3 (\to a + \to b) (\to b + \to c) 2 (\to c + \to a)] = 3 (\to a + \to b) \cdot [(\to b + \to c) \times 2 (\to c + \to a)]$
$= 6 (\to a + \to b) \cdot [(\to b \times \to c) + (\to b \times \to a) + (\to c \times \to c) + (\to c \times \to a)]$
$= 6 [\to a \to b \to c] + 6 [\to a \to b \to c]$
$= 12 [\to a \to b \to c] = 60$

$(C)$
$\frac{1}{2} ∣ ∣ ∣ \to a \times \to b ∣ ∣ ∣ = 20$
$∴ \frac{1}{2} ∣ ∣ ∣ (2 \to a + 3 \to b) \times (\to a - \to b) ∣ ∣ ∣ = \frac{1}{2} ∣ ∣ ∣ - 2 (\to a \times \to b) + 3 (\to b \times \to a) ∣ ∣ ∣$
$= \frac{1}{2} ∣ ∣ ∣ 5 (\to b \times \to a) ∣ ∣ ∣$
$= 5 \times 20 = 100$

$(D)$
$∣ ∣ ∣ \to a \times \to b ∣ ∣ ∣ = 30$
$∴ ∣ ∣ ∣ (\to a + \to b) \times \to a ∣ ∣ ∣ = ∣ ∣ ∣ \to a \times \to a + \to b \times \to a ∣ ∣ ∣$
$= 30$

Suggest Corrections

Similar questions

Q. Match

List I

with the

List II

and select the correct answer using the code given below the lists :

\begin{matrix} List I & List II (A) & Area of a triangle with adjacent sides determined by vectors \to a and \to b is 1 . Then the area of & (P) & 6 the triangle with adjacent sides determined by (3 \to a + 4 \to b) and (\to a - 3 \to b) is (B) & Volume of parallelopiped determined by vectors \to a, \to b, \to c is \frac{1}{4} . Then the volume of the & (Q) & 9 parallelopiped determined by vectors 3 (\to a + \to b), (\to b + \to c), 4 (\to c + \to a) is (C) & Area of a parallelogram with adjacent sides determined by vectors \to a and \to b is 8 . Then the & (R) & 13 area of the parallelogram with adjacent sides determined by vectors (2 \to a - \to b) and \to b is (D) & Volume of tetrahedron determined by vectors \to a, \to b and \to c is \frac{1}{2} . Then the volume of the & (S) & 16 tetrahedron determined by vectors 2 (\to a \times \to b), 3 (\to b \times \to c) and (\to c \times \to a) is \end{matrix}

Which of the following is the only CORRECT combination?

Q. Match

List I

with the

List II

and select the correct answer using the code given below the lists :

\begin{matrix} List I & List II (A) & Area of a triangle with adjacent sides determined by vectors \to a and \to b is 1 . Then the area of & (P) & 6 the triangle with adjacent sides determined by (3 \to a + 4 \to b) and (\to a - 3 \to b) is (B) & Volume of parallelopiped determined by vectors \to a, \to b, \to c is \frac{1}{4} . Then the volume of the & (Q) & 9 parallelopiped determined by vectors 3 (\to a + \to b), (\to b + \to c), 4 (\to c + \to a) is (C) & Area of a parallelogram with adjacent sides determined by vectors \to a and \to b is 8 . Then the & (R) & 13 area of the parallelogram with adjacent sides determined by vectors (2 \to a - \to b) and \to b is (D) & Volume of tetrahedron determined by vectors \to a, \to b and \to c is \frac{1}{2} . Then the volume of the & (S) & 16 tetrahedron determined by vectors 2 (\to a \times \to b), 3 (\to b \times \to c) and (\to c \times \to a) is \end{matrix}

Which of the following is the only CORRECT combination?

Q. Volume of parallelopiped formed by vectors

\to a \times \to b, \to b \times \to c

and

\to c \times \to a

36

cubic units. Based on the given information above match the following by appropriately matching the lists given in Column I and Column II.

\begin{matrix} C o l u m n 1 & C o l u m n 2 a. Volume of parallelopiped formed by vectors & p. 0 cubic units \to a, \to b and \to c is b. Volume of tetrahedron formed by vectors & q. 12 cubic units \to a, \to b and \to c is c. Volume of parallelopiped formed by vectors & r. 6 cubic units \to a + \to b, \to b + \to c and \to c + \to a is d. Volume of parallelopiped formed by vectors & s. 1 cubic unit \to a - \to b, \to b - \to c and \to c - \to a is \end{matrix}

If $\to a$ and $\to b$ are two vectors such that $| \to a | = 3, | \to b | = 2$ and angle between $\to a$ and $\to b is \frac{π}{3},$ then the area of the triangle with adjacent sides $\to a + 2 \to b$ and $2 \to a + \to b$ in sq. units is