JEE MAIN - Mathematics (2024 - 5th April Morning Shift - No. 28)
Let $$\overrightarrow{\mathrm{a}}=\hat{i}-3 \hat{j}+7 \hat{k}, \overrightarrow{\mathrm{b}}=2 \hat{i}-\hat{j}+\hat{k}$$ and $$\overrightarrow{\mathrm{c}}$$ be a vector such that $$(\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}) \times \overrightarrow{\mathrm{c}}=3(\overrightarrow{\mathrm{c}} \times \overrightarrow{\mathrm{a}})$$.
If $$\vec{a} \cdot \vec{c}=130$$, then $$\vec{b} \cdot \vec{c}$$ is equal to __________.
Answer
30
Explanation
$$(\vec{a}+2 \vec{b}) \times \vec{c}=3(\vec{c} \times \vec{a})$$
$$\begin{aligned} \Rightarrow \quad & \vec{b} \times \vec{c}+2(\vec{a} \times \vec{c})=0 \\ & (\vec{b}+2 \vec{a}) \times \vec{c}=0 \\ & \vec{c}=\lambda(\vec{b}+2 \vec{a}) \\ & \vec{c} \cdot \vec{a}=130 \Rightarrow \lambda=1 \\ & \vec{c}=4 \hat{i}-7 \hat{j}+15 \hat{k} \\ & \vec{b} . \vec{c}=30 \end{aligned}$$
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