Question

Show that the direction cosines of a vector equally inclined to the axes $OX,OY$ and $OZ$ are $\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}}$.

Answer 1

Let vector be $r=ai+bj+ck$

Dr's eye $a,b,c$

since are equally inclined

$\frac{a}{magofr}=\frac{b}{magofr}=\frac{c}{magofr}$

$a=b=c$

$\therefore$ vector is $\overline{r}=ai+aj+ak$

magnitude of $\overline{r}=\sqrt{a^2+a^2+a^2}$

$|\overline{r}|=\sqrt{3a^2}=\sqrt{3}a$

Dc's are $(\frac{a}{\sqrt{3}a},\frac{b}{\sqrt{3}a},\frac{c}{\sqrt{3}a})$

$=(\frac{a}{\sqrt{3}a},\frac{a}{\sqrt{3}a},\frac{a}{\sqrt{3}a})$

$=(\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}})$

Hence proved.