If the vectors $\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \lambda \hat{\mathbf{i}}-4…
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
$5$
$-5$
$8$
$-8$
Solution
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}
$