If the projection of $\overrightarrow{\mathrm{a}}=\widehat{\mathrm{ı}}-2\widehat{\mathrm{ȷ}}+3\widehat{\mathrm{k}}\text{ on }\overrightarrow{\mathrm{b}}=\widehat{2}\mathrm{l}+\mathrm{\lambda }\widehat{\mathrm{k}}$ is zero, then the value of 𝜆 is

Given, $\overrightarrow{\mathrm{a}}=\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+3\widehat{\mathrm{k}}\text{ and }\overrightarrow{\mathrm{b}}=2\widehat{\mathrm{i}}+\mathrm{\lambda }\widehat{\mathrm{k}}$
$\therefore \mathrm{Projection} \mathrm{of} \overrightarrow{\mathrm{a}}\text{ on }\overrightarrow{\mathrm{b}}=\frac{(\overrightarrow{\mathrm{a}}\cdot \overrightarrow{\mathrm{b}})}{\left|\overrightarrow{\mathrm{b}}\right|}$
$\begin{array}{l}=\frac{(\widehat{\mathrm{i}}-2\widehat{\mathrm{j}}+3\widehat{\mathrm{k}})\cdot (2\widehat{\mathrm{i}}+\mathrm{\lambda }\widehat{\mathrm{k}})}{|2\widehat{\mathrm{i}}+\mathrm{\lambda }\widehat{\mathrm{k}}|}\\ =\frac{2+3\mathrm{\lambda }}{\sqrt{2^2+{\mathrm{\lambda }}^2}}\\ =\frac{2+3\mathrm{\lambda }}{\sqrt{4+{\mathrm{\lambda }}^2}}\end{array}$
$\therefore \frac{2+3\mathrm{\lambda }}{\sqrt{4+{\mathrm{\lambda }}^2}}=0[\because \text{ Projection of }\overrightarrow{\mathrm{a}}\text{ on }\overrightarrow{\mathrm{b}}=0]$
$\begin{array}{r}\Rightarrow 3\mathrm{\lambda }+2=0\\ \Rightarrow \mathrm{\lambda }=-\frac{2}{3}\end{array}$