Mathematics - Vector Question with Solution | TestHub

Let $\overline{\mathrm{a}}={\mathrm{a}}_1\hat{\mathrm{i}}+{\mathrm{a}}_2\hat{\mathrm{j}}+{\mathrm{a}}_3\hat{\mathrm{k}}$
${\mathrm{a}}_1^2+{\mathrm{a}}_2^2+{\mathrm{a}}_3^2=1$
Let $\overline{\mathrm{b}}=2\overrightarrow{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}},\overline{\mathrm{c}}=\overrightarrow{\mathrm{i}}-2\hat{\mathrm{j}}-2\hat{\mathrm{k}}$
$\overline{\mathrm{d}}=\hat{\mathrm{k}}$
$\frac{\overline{\mathrm{a}}\cdot \overline{\mathrm{b}}}{|\mathrm{b}|}=\frac{\overline{\mathrm{a}}\cdot \overline{\mathrm{c}}}{|\mathrm{c}|}=\frac{\overline{\mathrm{a}}\cdot \overline{\mathrm{d}}}{|\mathrm{d}|}$
$\frac{2{\mathrm{a}}_1-{\mathrm{a}}_2+2{\mathrm{a}}_3}{3}=\frac{{\mathrm{a}}_1+2{\mathrm{a}}_2-2{\mathrm{a}}_3}{3}={\mathrm{a}}_3$
By solving
${\mathrm{a}}_1=\frac{7}{\sqrt{155}},{\mathrm{a}}_2=\frac{9}{\sqrt{155}},{\mathrm{a}}_3=\frac{5}{\sqrt{155}}$