JEE MAIN - Mathematics (2017 - 9th April Morning Slot - No. 12)
If the vector $$\overrightarrow b = 3\widehat j + 4\widehat k$$ is written as the
sum of a vector $$\overrightarrow {{b_1}} ,$$ paralel to $$\overrightarrow a = \widehat i + \widehat j$$ and a vector $$\overrightarrow {{b_2}} ,$$ perpendicular to $$\overrightarrow a ,$$ then $$\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} $$ is equal to :
$$ - 3\widehat i + 3\widehat j - 9\widehat k$$
$$6\widehat i - 6\widehat j + {9 \over 2}\widehat k$$
$$ - 6\widehat i + 6\widehat j - {9 \over 2}\widehat k$$
$$3\widehat i - 3\widehat j + 9\widehat k$$
Explanation
$$\overrightarrow {{b_1}} = {{\left( {\overrightarrow {{b_1}} .\overrightarrow a } \right)\widehat a} \over 1}$$
= $$\left\{ {{{\left( {3\widehat j + 4\widehat k} \right).\left( {\widehat i + \widehat j} \right)} \over {\sqrt 2 }}} \right\}\left( {{{\widehat i + \widehat j} \over {\sqrt 2 }}} \right)$$
= $${{3\left( {\widehat i + \widehat j} \right)} \over {\sqrt 2 \times \sqrt 2 }} = {{3\left( {\widehat i + \widehat j} \right)} \over 2}$$
$$\overrightarrow {{b_1}} + \overrightarrow {{b_2}} = \overrightarrow b $$
$$ \Rightarrow $$ $$\overrightarrow {{b_2}} = \overrightarrow b - \overrightarrow {{b_1}} $$
= $$\left( {3\widehat j + 4\widehat k} \right) - {3 \over 2}\left( {\widehat i + \widehat j} \right)$$
$$ \Rightarrow $$ $$\overrightarrow {{b_2}} $$ = $$ - {3 \over 2}\widehat i + {3 \over 2}\widehat j + 4\widehat k$$
& $$\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr {{3 \over 2}} & {{3 \over 2}} & 0 \cr { - {3 \over 2}} & {{3 \over 2}} & 4 \cr } } \right|$$
$$ \Rightarrow $$ $$\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} = \widehat i\left( 6 \right) - \widehat j\left( 6 \right) + \widehat k\left( { - {9 \over 4} + {9 \over 4}} \right)$$
$$ \Rightarrow $$ $$6\widehat i - 6\widehat j + {9 \over 2}\widehat k$$
= $$\left\{ {{{\left( {3\widehat j + 4\widehat k} \right).\left( {\widehat i + \widehat j} \right)} \over {\sqrt 2 }}} \right\}\left( {{{\widehat i + \widehat j} \over {\sqrt 2 }}} \right)$$
= $${{3\left( {\widehat i + \widehat j} \right)} \over {\sqrt 2 \times \sqrt 2 }} = {{3\left( {\widehat i + \widehat j} \right)} \over 2}$$
$$\overrightarrow {{b_1}} + \overrightarrow {{b_2}} = \overrightarrow b $$
$$ \Rightarrow $$ $$\overrightarrow {{b_2}} = \overrightarrow b - \overrightarrow {{b_1}} $$
= $$\left( {3\widehat j + 4\widehat k} \right) - {3 \over 2}\left( {\widehat i + \widehat j} \right)$$
$$ \Rightarrow $$ $$\overrightarrow {{b_2}} $$ = $$ - {3 \over 2}\widehat i + {3 \over 2}\widehat j + 4\widehat k$$
& $$\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} = \left| {\matrix{ {\widehat i} & {\widehat j} & {\widehat k} \cr {{3 \over 2}} & {{3 \over 2}} & 0 \cr { - {3 \over 2}} & {{3 \over 2}} & 4 \cr } } \right|$$
$$ \Rightarrow $$ $$\overrightarrow {{b_1}} \times \overrightarrow {{b_2}} = \widehat i\left( 6 \right) - \widehat j\left( 6 \right) + \widehat k\left( { - {9 \over 4} + {9 \over 4}} \right)$$
$$ \Rightarrow $$ $$6\widehat i - 6\widehat j + {9 \over 2}\widehat k$$
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