JEE MAIN - Mathematics (2022 - 29th June Evening Shift - No. 12)

Let  $$\overrightarrow a = \widehat i - 2\widehat j + 3\widehat k$$,   $$\overrightarrow b = \widehat i + \widehat j + \widehat k$$   and   $$\overrightarrow c $$   be a vector such that   $$\overrightarrow a + \left( {\overrightarrow b \times \overrightarrow c } \right) = \overrightarrow 0 $$   and   $$\overrightarrow b \,.\,\overrightarrow c = 5$$. Then the value of   $$3\left( {\overrightarrow c \,.\,\overrightarrow a } \right)$$   is equal to _________.
Answer
BONUS

Explanation

$$ \vec{a} \cdot \vec{b}=(\hat{i}-2 \hat{j}+3 \hat{k}) \cdot(\hat{i}+\hat{j}+\hat{k})=2 $$ ........(i)

Given: $\vec{a}+(\vec{b} \times \vec{c})=0$

$$ \Rightarrow \vec{a} \cdot \vec{b}=0 $$ ........(ii)

Equation (i) and equation (ii) are contradicting.

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