JEE MAIN - Mathematics (2021 - 27th July Morning Shift - No. 2)
Let $$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$$ and $$\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k$$. Then the vector product $$\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {\left( {\overrightarrow a \times \left( {\left( {\overrightarrow a - \overrightarrow b } \right) \times \overrightarrow b } \right)} \right) \times \overrightarrow b } \right)$$ is equal to :
$$5(34\widehat i - 5\widehat j + 3\widehat k)$$
$$7(34\widehat i - 5\widehat j + 3\widehat k)$$
$$7(30\widehat i - 5\widehat j + 7\widehat k)$$
$$5(30\widehat i - 5\widehat j + 7\widehat k)$$
Explanation
$$\overrightarrow a = \widehat i + \widehat j + 2\widehat k$$
$$\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k$$
$$\overrightarrow a + \overrightarrow b = 3\widehat j + 5\widehat k;\overrightarrow a.\overrightarrow b = - 1 + 2 + 6 = 7$$
$$\left( {\left( {\overrightarrow a \times \left( {\left( {\overrightarrow a - \overrightarrow b } \right) \times \overrightarrow b } \right)} \right) \times \overrightarrow b } \right)$$
$$\left( {\left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b - \overrightarrow b \times \overrightarrow b } \right)} \right) \times \overrightarrow b } \right)$$
$$\left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b - 0} \right)} \right) \times \overrightarrow b $$
$$\left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b } \right)} \right) \times \overrightarrow b $$
$$\left( {\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow a - \left( {\overrightarrow a .\overrightarrow a } \right)\overrightarrow b } \right) \times \overrightarrow b $$
$$\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow a \times \overrightarrow b - \left( {\overrightarrow a .\overrightarrow a } \right)\left( {\overrightarrow b \times \overrightarrow b } \right)$$
$$\left( {\overrightarrow a .\overrightarrow b } \right)\left( {\overrightarrow a \times \overrightarrow b } \right)$$
$$\overrightarrow a \times \overrightarrow b = \left| {\matrix{ i & j & k \cr 1 & 1 & 2 \cr { - 1} & 2 & 3 \cr } } \right| = - \widehat i - 5\widehat j + 3\widehat k$$
$$\therefore$$ $$7\left( { - \widehat i - 5\widehat j + 3\widehat k} \right)$$
$$\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {7\left( { - \widehat i - 5\widehat j + 3\widehat k} \right)} \right)$$
$$7\left( {0\widehat i + 3\widehat j + 5\widehat k} \right) \times \left( { - \widehat i - 5\widehat j + 3\widehat k} \right)$$
$$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 0 & 3 & 5 \cr { - 1} & { - 5} & 3 \cr } } \right|$$
$$ \Rightarrow 34\widehat i - (5)\widehat j + (3\widehat k)$$
$$ \Rightarrow 34\widehat i - 5\widehat j + 3\widehat k$$
$$\therefore$$ $$7\left( {0\widehat i + 3\widehat j + 5\widehat k} \right) \times \left( { - \widehat i - 5\widehat j + 3\widehat k} \right)$$
= $$7(34\widehat i - 5\widehat j + 3\widehat k)$$
$$\overrightarrow b = - \widehat i + 2\widehat j + 3\widehat k$$
$$\overrightarrow a + \overrightarrow b = 3\widehat j + 5\widehat k;\overrightarrow a.\overrightarrow b = - 1 + 2 + 6 = 7$$
$$\left( {\left( {\overrightarrow a \times \left( {\left( {\overrightarrow a - \overrightarrow b } \right) \times \overrightarrow b } \right)} \right) \times \overrightarrow b } \right)$$
$$\left( {\left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b - \overrightarrow b \times \overrightarrow b } \right)} \right) \times \overrightarrow b } \right)$$
$$\left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b - 0} \right)} \right) \times \overrightarrow b $$
$$\left( {\overrightarrow a \times \left( {\overrightarrow a \times \overrightarrow b } \right)} \right) \times \overrightarrow b $$
$$\left( {\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow a - \left( {\overrightarrow a .\overrightarrow a } \right)\overrightarrow b } \right) \times \overrightarrow b $$
$$\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow a \times \overrightarrow b - \left( {\overrightarrow a .\overrightarrow a } \right)\left( {\overrightarrow b \times \overrightarrow b } \right)$$
$$\left( {\overrightarrow a .\overrightarrow b } \right)\left( {\overrightarrow a \times \overrightarrow b } \right)$$
$$\overrightarrow a \times \overrightarrow b = \left| {\matrix{ i & j & k \cr 1 & 1 & 2 \cr { - 1} & 2 & 3 \cr } } \right| = - \widehat i - 5\widehat j + 3\widehat k$$
$$\therefore$$ $$7\left( { - \widehat i - 5\widehat j + 3\widehat k} \right)$$
$$\left( {\overrightarrow a + \overrightarrow b } \right) \times \left( {7\left( { - \widehat i - 5\widehat j + 3\widehat k} \right)} \right)$$
$$7\left( {0\widehat i + 3\widehat j + 5\widehat k} \right) \times \left( { - \widehat i - 5\widehat j + 3\widehat k} \right)$$
$$\left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr 0 & 3 & 5 \cr { - 1} & { - 5} & 3 \cr } } \right|$$
$$ \Rightarrow 34\widehat i - (5)\widehat j + (3\widehat k)$$
$$ \Rightarrow 34\widehat i - 5\widehat j + 3\widehat k$$
$$\therefore$$ $$7\left( {0\widehat i + 3\widehat j + 5\widehat k} \right) \times \left( { - \widehat i - 5\widehat j + 3\widehat k} \right)$$
= $$7(34\widehat i - 5\widehat j + 3\widehat k)$$
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