Question

A vector of magnitude 21 having direction ratios proportional to 2, –3, 6 is ____________.

Answer 1

Let us suppose vector is given by $\overrightarrow{r}=a\hat{i}+b\hat{j}+c\hat{k}$
$$\begin{gathered} \mathrm{Since}\frac{a}{2}=\frac{b}{-3}=\frac{c}{6}=h\left(\mathrm{say}\right) \\ \mathrm{Then}\overrightarrow{r}=2h\hat{i}-3h\hat{j}+6h\hat{k} \\ =2h\hat{i}-3h\hat{j}+6h\hat{k} \\ \mathrm{i}.\mathrm{e}\overrightarrow{r}=h\left(2\hat{i}-3\hat{j}+6\hat{k}\right)\mathrm{and}\left|\overrightarrow{r}\right|=21 \\ \therefore 4h^2+9h^2+36h^2={\left(21\right)}^2 \\ \mathrm{i}.\mathrm{e}49h^2=21\times 21 \\ \mathrm{i}.\mathrm{e}{h}^2=3\times 3 \\ \mathrm{i}.\mathrm{e}h=3 \\ \therefore \mathrm{vector}\overrightarrow{r}=6\hat{i}-9\hat{j}+18\hat{k} \end{gathered}$$