JEE MAIN - Mathematics (2023 - 29th January Evening Shift - No. 4)
If $$\overrightarrow a = \widehat i + 2\widehat k,\overrightarrow b = \widehat i + \widehat j + \widehat k,\overrightarrow c = 7\widehat i - 3\widehat j + 4\widehat k,\overrightarrow r \times \overrightarrow b + \overrightarrow b \times \overrightarrow c = \overrightarrow 0 $$ and $$\overrightarrow r \,.\,\overrightarrow a = 0$$. Then $$\overrightarrow r \,.\,\overrightarrow c $$ is equal to :
36
30
34
32
Explanation
$$(\overrightarrow r - \overrightarrow c ) \times \overrightarrow b = 0$$
$$\overrightarrow r = \lambda \overrightarrow b + \overrightarrow c $$
$$ \Rightarrow \lambda \overrightarrow b \,.\,\overrightarrow a + \overrightarrow c \,.\,\overrightarrow a = 0$$
$$ \Rightarrow \lambda (3) + (7 + 8) = 0$$
$$ \Rightarrow \lambda = - 5$$
$$\overrightarrow r = 5\overrightarrow b + \overrightarrow c $$
$$ = - 5\widehat i - 5\widehat j - 5\widehat k + (7\widehat i + 3\widehat j + 4\widehat k)$$
$$ = 2\widehat i - 8\widehat j - \widehat k$$
$$\therefore$$ $$\overrightarrow r \,.\,\overrightarrow c = 17 + 24 - 4 = 34$$
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