JEE MAIN - Mathematics (2022 - 27th June Morning Shift - No. 13)
Let $$\overrightarrow a = \widehat i + \widehat j - \widehat k$$ and $$\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$$. Then the number of vectors $$\overrightarrow b $$ such that $$\overrightarrow b \times \overrightarrow c = \overrightarrow a $$ and $$|\overrightarrow b | \in $$ {1, 2, ........, 10} is :
0
1
2
3
Explanation
$$\overrightarrow a = \widehat i + \widehat j - \widehat k$$
$$\overrightarrow c = 2\widehat i - 3\widehat j + 2\widehat k$$
Now, $$\overrightarrow b \times \overrightarrow c = \overrightarrow a $$
$$\overrightarrow c \,.\,(\overrightarrow b \times \overrightarrow c ) = \overrightarrow c \,.\,\overrightarrow a $$
$$\overrightarrow c \,.\,\overrightarrow a = 0$$
$$ \Rightarrow (\widehat i + \widehat j - \widehat k)(2\widehat i - 3\widehat j + 2\widehat k) = 0$$
$$ = 2 - 3 - 2 = 0$$
$$ \Rightarrow - 3 = 0$$ (Not possible)
$$\Rightarrow$$ No possible value of $$\overrightarrow b $$ is possible.
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