JEE Advance - Mathematics (2011 - Paper 2 Offline - No. 12)
Let $$\overrightarrow a = - \widehat i - \widehat k,\overrightarrow b = - \widehat i + \widehat j$$ and $$\overrightarrow c = \widehat i + 2\widehat j + 3\widehat k$$ be three given vectors. If $$\overrightarrow r $$ is a vector such that $$\overrightarrow r \times \overrightarrow b = \overrightarrow c \times \overrightarrow b $$ and $$\overrightarrow r .\overrightarrow a = 0,$$ then the value of $$\overrightarrow r .\overrightarrow b $$ is
Answer
9
Explanation
Since it is given that $$\overrightarrow r \times \overrightarrow b = \overrightarrow c \times \overrightarrow b $$, taking cross product with
$$\overrightarrow a \times (\overrightarrow r \times \overrightarrow b ) = \overrightarrow a \times (\overrightarrow c \times \overrightarrow b )$$
$$(\overrightarrow a \,.\,\overrightarrow b )\overrightarrow r - (\overrightarrow a \,.\,\overrightarrow r )\overrightarrow b = \overrightarrow a \times (\overrightarrow c \times \overrightarrow b )$$
$$ \Rightarrow \overrightarrow r = - 3\widehat i + 6\widehat j + 3\widehat k$$
$$\overrightarrow r \,.\,\overrightarrow b = 3 + 6 = 9$$
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