If a¯=2i^+j^−3k^,b¯=i^−2j^+3k^,c¯=−i^+j^−4k^ and d¯=i^+j^+2k^ then (a¯×b¯)×(c¯×d¯)=

If a×b=i j k2 1 −31 −2 3=i(3−6)−j(6+3)+k(−4−1)=−3i−aj−5kand c×d= i j k−1 1 −4 1 1 2=i(2+4)−j(−2+4)+k(−1−1)=6i−2j−2k(a×b)×(c×d)= i j k−3 −9 −5 6 −2 −2 =i(18−10)−j(6+30)+k(6+54)=8i−36j+60k

If a¯=2i^+j^−3k^,b¯=i^−2j^+3k^,c¯=−i^+j^−4k^ and d¯=i^+j^+2k^ then (a¯×b¯)×(c¯×d¯)=

If a×b=i j k2 1 −31 −2 3=i(3−6)−j(6+3)+k(−4−1)=−3i−aj−5kand c×d= i j k−1 1 −4 1 1 2=i(2+4)−j(−2+4)+k(−1−1)=6i−2j−2k(a×b)×(c×d)= i j k−3 −9 −5 6 −2 −2 =i(18−10)−j(6+30)+k(6+54)=8i−36j+60k

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If $\bar{a} = 2 \hat{i} + \hat{j} - 3 \hat{k}, \bar{b} = \hat{i} - 2 \hat{j} + 3 \hat{k},$ $\bar{c} = - \hat{i} + \hat{j} - 4 \hat{k}$ and $\bar{d} = \hat{i} + \hat{j} + 2 \hat{k}$ then $(\bar{a} \times \bar{b}) \times (\bar{c} \times \bar{d}) =$

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$- 7 \hat{i} + \hat{j} + 3 \hat{k}$

$8 \hat{i} - 36 \hat{j} + 60 \hat{k}$

$5 \hat{i} + \hat{j} - \hat{k}$

$- 8 \hat{i} - 36 \hat{j} + 12 \hat{k}$

answer is B.

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Detailed Solution

If $a \times b = |\begin{array}{l} i j k \\ 2 1 - 3 \\ 1 - 2 3 \end{array}|$
$= i (3 - 6) - j (6 + 3) + k (- 4 - 1) = - 3 i - a j - 5 k$
$a n d c \times d = |\begin{array}{l} i j k \\ - 1 1 - 4 \\ 1 1 2 \end{array}|$
$\begin{array}{l} = i (2 + 4) - j (- 2 + 4) + k (- 1 - 1) = 6 i - 2 j - 2 k \\ (a \times b) \times (c \times d) = |\begin{array}{l} i j k \\ - 3 - 9 - 5 \\ 6 - 2 - 2 \end{array}| \\ = i (18 - 10) - j (6 + 30) + k (6 + 54) \\ = 8 i - 36 j + 60 k \end{array}$