Question

For what value of $\lambda$, the vectors $\overrightarrow{a}=2\overrightarrow{i}+\lambda \overrightarrow{j}+\overrightarrow{k}$ and $\overrightarrow{b}=-\overrightarrow{i}+2\overrightarrow{j}-3\overrightarrow{k}$ are orthogonal to each other ?

Answer 1

$\overrightarrow{a}=2\hat{i}+\lambda \hat{j}+\hat{k}$

$\overrightarrow{b}=-\hat{i}+2\hat{j}-3\hat{k}$

are orthogonal.

$\overrightarrow{a}.\overrightarrow{b}=0$

$-2+2\lambda -3=0$

$2\lambda =5$

$\lambda =\frac{5}{2}$