Volume of parallelepiped formed by vectors a→×b→,b→×c→ and c→×a→ is 36 sq. units.

P → a → b → c → = 2 ⇒ [ 2 a → × b → 3 b → × c → c → × a → ] = 6 [ a → b → c → ] 2 = 6 × 4 = 24 Q → [ a → b → c → ] = 5 ⇒ 6 [ a → + b → b → + c → c → + a → ] = 12 [ a → b → c → ] = 60 ⇒ R → 1 2 | a → × b → | = 20 ⇒ 1 2 ∣ ( 2 a → + 3 b → ) × ( a → − b → ) ⇒ 1 2 | − 2 ( a → × b → ) − 3 ( a → × b → ) | ⇒ 5 2 × 40 = 100 ⇒ S → | a → × b → | = 30 ⇒ | ( a → + b → ) × a → | = | b → × a → | = 30

Volume of parallelepiped formed by vectors, a → , b → and c → is , Volume of tetrahedron formed by vectors a → , b → and c → is , Volume of parallelepiped formed by vectors a → + b → , b → + c → and c → + a → is , Volume of parallelepiped formed by vectors a → − b → , b → − c → and c → − a → is

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Volume of parallelepiped formed by vectors $\vec{a} \times \vec{b}, \vec{b} \times \vec{c}$ and $\vec{c} \times \vec{a}$ is 36 sq. units.

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Volume of parallelepiped formed by vectors, $\vec{a}, \vec{b}$ and $\vec{c}$ is

Volume of tetrahedron formed by vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is

Volume of parallelepiped formed by vectors $\vec{a} + \vec{b}, \vec{b} + \vec{c}$ and $\vec{c} + \vec{a}$ is

Volume of parallelepiped formed by vectors $\vec{a} - \vec{b}, \vec{b} - \vec{c}$ and $\vec{c} - \vec{a}$ is

0 sq. units

12 sq. units

6 sq. units

1 sq. units

answer is D, C, B, A.

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Detailed Solution

$P \to [\begin{array}{lll} \vec{a} \vec{b} \vec{c} \end{array}] = 2$

$\Rightarrow [2 \vec{a} \times \vec{b} 3 \vec{b} \times \vec{c} \vec{c} \times \vec{a}] = 6 [\vec{a} \vec{b} \vec{c}]^{2} = 6 \times 4 = 24$

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

$\Rightarrow 6 [\vec{a} + \vec{b} \vec{b} + \vec{c} \vec{c} + \vec{a}] = 12 [\vec{a} \vec{b} \vec{c}] = 60$

$\Rightarrow R \to \frac{1}{2} | \vec{a} \times \vec{b} | = 20 \Rightarrow \frac{1}{2} ∣ (2 \vec{a} + 3 \vec{b}) \times (\vec{a} - \vec{b})$

$\Rightarrow \frac{1}{2} | - 2 (\vec{a} \times \vec{b}) - 3 (\vec{a} \times \vec{b}) | \Rightarrow \frac{5}{2} \times 40 = 100$

$\Rightarrow S \to | \vec{a} \times \vec{b} | = 30 \Rightarrow | (\vec{a} + \vec{b}) \times \vec{a} | = | \vec{b} \times \vec{a} | = 30$

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