The value of  $\lambda$ for which two vectors $\mathbf{a}=5\hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}\text{ and }\mathbf{b}=\hat{\mathbf{i}}-2\hat{\mathbf{j}}+\hat{\mathbf{k}}$ are perpendicular  to. each other is 

For two vectors a and b to be perpendicular, a · b = 0
Thus, $(5\hat{\mathbf{i}}+\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}})·(\hat{\mathbf{i}}-2\hat{\mathbf{j}}+\hat{\mathbf{k}})=0$

$5(\hat{\mathbf{i}}·\hat{\mathbf{i}})-2\lambda (\hat{\mathbf{j}}·\hat{\mathbf{j}})+1(\hat{\mathbf{k}}·\hat{\mathbf{k}})=0$

$\Rightarrow 5-2\lambda +1=0$

$\Rightarrow$  $6-2\lambda =0\Rightarrow \lambda =3$