Match List-I with List-II and select the correct answer using the code given below the lists:List IList II(P) Volume of parallelepiped determined by vector is 2. Then the volume of the parallelepiped determined by vectors is(1) 100(Q) Volume of parallelepiped determined by vectors is 5. Then the volume of the parallelepiped determined by vectors is(2) 30(R) Area of a triangle with adjacent sides determined by vectors is 20. Then the area of the triangle with adjacent sides determined by vectors is(3) 24(S) Area of a parallelogram with adjacent sides determined by vectors is 30. Then the area of the parallelogram with adjacent sides determined by vectors is(4) 60

Engineering

Mathematics

Scalar Triple Product

Question

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

List I	List II
(P) Volume of parallelepiped determined by vector $\vec{a}, \vec{b} and \vec{c}$ is 2. Then the volume of the parallelepiped determined by vectors $2 (\vec{a} \times \vec{b}), 3 (\vec{b} \times \vec{c}) and (\vec{c} \times \vec{a})$ is	(1) 100
(Q) Volume of parallelepiped determined by vectors $\vec{a}, \vec{b} and \vec{c}$ is 5. Then the volume of the parallelepiped determined by vectors $3 (\vec{a} + \vec{b}), (\vec{b} + \vec{c}) and 2 (\vec{c} + \vec{a})$ is	(2) 30
(R) Area of a triangle with adjacent sides determined by vectors $\vec{a} and \vec{b}$ is 20. Then the area of the triangle with adjacent sides determined by vectors $(2 \vec{a} + 3 \vec{b}) and (\vec{a} - \vec{b})$ is	(3) 24
(S) Area of a parallelogram with adjacent sides determined by vectors $\vec{a} and \vec{b}$ is 30. Then the area of the parallelogram with adjacent sides determined by vectors $(\vec{a} + \vec{b}) and \vec{a}$ is	(4) 60

P → 4

Q → 2

R → 3

S → 1

P → 2

Q → 3

R → 1

S → 4

P → 3

Q → 4

R → 1

S → 2

P → 1

Q → 4

R → 3

S → 2

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$P : | [\vec{a} \vec{b} \vec{c}] | = 2$

$∴ {[\begin{matrix} 2 (\vec{a} \times \vec{b}) & 3 (\vec{b} \times \vec{c}) & (\vec{c} \times \vec{a}) \end{matrix}]}^{} = 6 {[\vec{a} \vec{b} \vec{c}]}^{2} = 24$

$Q:|[\vec{a} \vec{b} \vec{c}] | = 5$

$∴ | {[\begin{matrix} 3 (\vec{a} + \vec{b}) & \vec{b} + \vec{c} & 2 (\vec{c} + \vec{a}) \end{matrix}]}^{} | = 6 (z) {[\vec{a} \vec{b} \vec{c}]}^{} = 12 (5) = 60$

$R : \frac{1}{2} | \vec{a} \times \vec{b} | = 20 \Rightarrow | \vec{a} \times \vec{b} | = 40$

$∴ Δ = \frac{1}{2} | (2 \vec{a} + 3 \vec{b}) \times (\vec{a} - \vec{b}) | = \frac{1}{2} | - 2 \vec{a} \times \vec{b} + 3 \vec{b} \times \vec{a} | = \frac{5}{2} | \vec{a} \times \vec{b} |$

$S : | \vec{a} \times \vec{b} | = 30$

$| (\vec{a} \times \vec{b}) \times \vec{a} | = | \vec{b} \times \vec{a} | = 30$