Let a→=i^+j^+2k^ and b→=−i^+2j^+3k^ then the vector product (a¯+b¯)×((a¯×((a¯−b¯)×b¯))×b¯) is ________

a→=i^+j^+2k^b→=−i^+2j^+3k^a→+b→=3i^+5j^;a→b→=−1+2+6=7((a→×((a→−b→)×b→))×b→)((a→×((a→×b→−b→×b→))×b→)(a→×((a→×b→−0))×b→(a→×((a→×b→))×b→((a→⋅b→)a→−(a→⋅a→)b→)×b→(a→⋅b→)a→×b→−(a→⋅a→)b→×b→(a→⋅b→)(a→×b→)a→×b→=ijk112−123=−i^−5j^+3k^∴7(−i^−5j^+3k^)(a→+b→)×(7(−i^−5j^+3k^))7(0i^+3j^+5k^)×(−i^−5j^+3k^)i^j^k^035−1−53⇒34i^−(5)j^+(3k^)⇒34i^−5j^+3k^∴7(34i^−5j^+3k^)

Let a→=i^+j^+2k^ and b→=−i^+2j^+3k^ then the vector product (a¯+b¯)×((a¯×((a¯−b¯)×b¯))×b¯) is ________

a→=i^+j^+2k^b→=−i^+2j^+3k^a→+b→=3i^+5j^;a→b→=−1+2+6=7((a→×((a→−b→)×b→))×b→)((a→×((a→×b→−b→×b→))×b→)(a→×((a→×b→−0))×b→(a→×((a→×b→))×b→((a→⋅b→)a→−(a→⋅a→)b→)×b→(a→⋅b→)a→×b→−(a→⋅a→)b→×b→(a→⋅b→)(a→×b→)a→×b→=ijk112−123=−i^−5j^+3k^∴7(−i^−5j^+3k^)(a→+b→)×(7(−i^−5j^+3k^))7(0i^+3j^+5k^)×(−i^−5j^+3k^)i^j^k^035−1−53⇒34i^−(5)j^+(3k^)⇒34i^−5j^+3k^∴7(34i^−5j^+3k^)

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Let $\vec{a} = \hat{i} + \hat{j} + 2 \hat{k}$ and $\vec{b} = - \hat{i} + 2 \hat{j} + 3 \hat{k}$ then the vector product $(\bar{a} + \bar{b}) \times ((\bar{a} \times ((\bar{a} - \bar{b}) \times \bar{b})) \times \bar{b})$ is ________

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$(30 \hat{i} - 5 \hat{j} + 7 \hat{k})$

$6 (34 \hat{i} - 5 \hat{j} + 3 \hat{k})$

$7 (34 \hat{i} - 5 \hat{j} + 3 \hat{k})$

$7 (30 \hat{i} - 5 \hat{j} + 7 \hat{k})$

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Detailed Solution

$\begin{aligned} \vec{a} = \hat{i} + \hat{j} + 2 \hat{k} \\ \vec{b} = - \hat{i} + 2 \hat{j} + 3 \hat{k} \\ \vec{a} + \vec{b} = 3 \hat{i} + 5 \hat{j}; \vec{a} \vec{b} = - 1 + 2 + 6 = 7 \\ ((\vec{a} \times ((\vec{a} - \vec{b}) \times \vec{b})) \times \vec{b}) \\ ((\vec{a} \times ((\vec{a} \times \vec{b} - \vec{b} \times \vec{b})) \times \vec{b}) \\ (\vec{a} \times ((\vec{a} \times \vec{b} - 0)) \times \vec{b} \\ (\vec{a} \times ((\vec{a} \times \vec{b})) \times \vec{b} \\ ((\vec{a} \cdot \vec{b}) \vec{a} - (\vec{a} \cdot \vec{a}) \vec{b}) \times \vec{b} \\ (\vec{a} \cdot \vec{b}) \vec{a} \times \vec{b} - (\vec{a} \cdot \vec{a}) \vec{b} \times \vec{b} \\ (\vec{a} \cdot \vec{b}) (\vec{a} \times \vec{b}) \end{aligned}$

$\begin{aligned} \vec{a} \times \vec{b} = |\begin{array}{ccc} i & j & k \\ 1 & 1 & 2 \\ - 1 & 2 & 3 \end{array}| = - \hat{i} - 5 \hat{j} + 3 \hat{k} \\ ∴ 7 (- \hat{i} - 5 \hat{j} + 3 \hat{k}) \\ (\vec{a} + \vec{b}) \times (7 (- \hat{i} - 5 \hat{j} + 3 \hat{k})) \\ 7 (0 \hat{i} + 3 \hat{j} + 5 \hat{k}) \times (- \hat{i} - 5 \hat{j} + 3 \hat{k}) \\ |\begin{array}{lrr} \hat{i} & \hat{j} & \hat{k} \\ 0 & 3 & 5 \\ - 1 & - 5 & 3 \end{array}| \\ \Rightarrow 34 \hat{i} - (5) \hat{j} + (3 \hat{k}) \\ \Rightarrow 34 \hat{i} - 5 \hat{j} + 3 \hat{k} \\ ∴ 7 (34 \hat{i} - 5 \hat{j} + 3 \hat{k}) \end{aligned}$