find the measure of acute angle between the vectors whose direction ratio are 4,2,3 and 1,2,4

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Please find below the solution to the asked query:

$$\begin{gathered} \mathrm{We} \mathrm{have} \\ \overrightarrow{\mathrm{a}}=4\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}} \\ \overrightarrow{\mathrm{b}}=\hat{\mathrm{i}}+2\hat{\mathrm{j}}+4\hat{\mathrm{k}} \\ \mathrm{If} \mathrm{\theta } \mathrm{is} \mathrm{required} \mathrm{acute} \mathrm{angle}, \mathrm{then} \\ \mathrm{cos\theta }=\frac{\left|\overrightarrow{\mathrm{a}}·\overrightarrow{\mathrm{b}}\right|}{\left|\overrightarrow{\mathrm{a}}\right|\left|\overrightarrow{\mathrm{b}}\right|}=\frac{\left(4\hat{\mathrm{i}}+2\hat{\mathrm{j}}+3\hat{\mathrm{k}}\right).\left(\hat{\mathrm{i}}+2\hat{\mathrm{j}}+4\hat{\mathrm{k}}\right)}{\sqrt{16+4+9}\sqrt{1+4+16}} \\ =\frac{4+4+12}{\sqrt{29}\sqrt{21}}=\frac{20}{\sqrt{29}\sqrt{21}} \\ \mathbf{\Rightarrow }\mathbf{\theta }\mathbf{=}{\mathbf{cos}}^{\mathbf{-}\mathbf{1}}\left\{\frac{\mathbf{20}}{\sqrt{\mathbf{29}}\sqrt{\mathbf{21}}}\right\} \end{gathered}$$

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