(a) If a and b are position vectors of point A and point B respectively, find the position vector of point C on BA produced such that BC =3 BA.

C divides BA in the ratio $\displaystyle 3: 2$ externally Required vector $\displaystyle =\overrightarrow{\mathrm{c}}=\frac{3 \vec{a}-2 \vec{b}}{3-2}=3 \vec{a}-2 \vec{b}$

Unit vector equally inclined along coordinate axes is $\displaystyle \frac{\hat{\mathrm{i}}}{\sqrt{3}}+\frac{\hat{\mathrm{j}}}{\sqrt{3}}+\frac{\hat{\mathrm{k}}}{\sqrt{3}} \vec{r}=5 \sqrt{3}\left(\frac{\hat{\mathrm{i}}}{\sqrt{3}}+\frac{\hat{\mathrm{j}}}{\sqrt{3}}+\frac{\hat{\mathrm{k}}}{\sqrt{3}}\right)=5 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}+5 \hat{\mathrm{k}} \quad($ or $\displaystyle -5 \hat{\mathrm{i}}-5 \hat{\mathrm{j}}-5 \hat{\mathrm{k}})$