Write the direction ratio's of the vector $\vec{a} = \hat{i} + \hat{j} - 2\hat{k}$ and hence calculate its direction cosines.

Direction Ratios: $(1, 1, -2)$; Direction Cosines: $(\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}})$

The correct answer is Option (2) → Direction Ratios: $(1, 1, -2)$; Direction Cosines: $(\frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}})$ ##

Note that the direction ratio's $a, b, c$ of a vector $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$ are just the respective components $x, y$ and $z$ of the vector. So, for the given vector, we have $a = 1, b = 1$ and $c = -2$. Further, if $l, m$ and $n$ are the direction cosines of the given vector, then

$l = \frac{a}{|\vec{r}|} = \frac{1}{\sqrt{6}}, \quad m = \frac{b}{|\vec{r}|} = \frac{1}{\sqrt{6}}, \quad n = \frac{c}{|\vec{r}|} = \frac{-2}{\sqrt{6}} \text{ as } |\vec{r}| = \sqrt{6}$

Thus, the direction cosines are $\left( \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}}, -\frac{2}{\sqrt{6}} \right)$.