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If $\vec{\beta}$ is perpendicular to both $\vec{\alpha}$ and $\vec{\gamma}$ where $\vec{\alpha}=\overrightarrow{\mathrm{k}}$ and
$\vec{\gamma}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$, then what is $\vec{\beta}$ equal to?
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$\vec{\gamma}=2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}+4 \hat{\mathrm{k}}$, then what is $\vec{\beta}$ equal to?
Solution:
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Verified Answer
The correct answer is:
$-3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}$
Let $\vec{\beta}=a \hat{i}+b \hat{j}+c \hat{k}$
since $\vec{\beta} \cdot \vec{\alpha}=0$ and $\vec{\beta} \cdot \vec{\gamma}=0$
$\therefore \quad c=0$ and $2 a+3 b=0 \Rightarrow a=-\frac{3 b}{2}$
Hence, $\vec{\beta}=-\frac{3 b}{2} \hat{i}+b \hat{j}=0 \Rightarrow-3 \hat{i}+2 \hat{j}=0$
since $\vec{\beta} \cdot \vec{\alpha}=0$ and $\vec{\beta} \cdot \vec{\gamma}=0$
$\therefore \quad c=0$ and $2 a+3 b=0 \Rightarrow a=-\frac{3 b}{2}$
Hence, $\vec{\beta}=-\frac{3 b}{2} \hat{i}+b \hat{j}=0 \Rightarrow-3 \hat{i}+2 \hat{j}=0$
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