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If the vectors $\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \lambda \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ are orthogonal to each other, then $\lambda$ is equal to
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Verified Answer
The correct answer is:
$8$
Let $\mathbf{a}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ and
$$
\mathbf{b}=\lambda \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+\hat{\mathbf{k}}
$$
Since, $\mathbf{a}$ and $\mathbf{b}$ are orthogonal to each other
$$
\begin{array}{lc}
\therefore & \mathbf{a} \cdot \mathbf{b}=0 \\
\Rightarrow & (\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \cdot(\lambda \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+\hat{\mathbf{k}})=0 \\
\Rightarrow & \lambda-12+4=0 \Rightarrow \lambda=8
\end{array}
$$
$$
\mathbf{b}=\lambda \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+\hat{\mathbf{k}}
$$
Since, $\mathbf{a}$ and $\mathbf{b}$ are orthogonal to each other
$$
\begin{array}{lc}
\therefore & \mathbf{a} \cdot \mathbf{b}=0 \\
\Rightarrow & (\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}) \cdot(\lambda \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+\hat{\mathbf{k}})=0 \\
\Rightarrow & \lambda-12+4=0 \Rightarrow \lambda=8
\end{array}
$$
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