Find the direction cosines of the vector $\hat i + 2\hat j + 3\hat k$.

The direction cosines of the vector $x\hat i + y\hat j + z\hat k$ are $< \cfrac{x}{{\sqrt {{x^2} + {y^2} + {z^2}} }},\cfrac{y}{{\sqrt {{x^2} + {y^2} + {z^2}} }},\cfrac{z}{{\sqrt {{x^2} + {y^2} + {z^2}} }} >$

Here, $\vec a = \hat i + 2\hat j + 3\hat k$ $\therefore$ $x = 1,y = 2,z = 3$
$\therefore$

Direction cosines of $\vec a$ are
$< \cfrac{1}{{\sqrt {1 + 4 + 9} }},\cfrac{2}{{\sqrt {1 + 4 + 9} }},\cfrac{3}{{\sqrt {1 + 4 + 9} }} >$
i.e., $< \cfrac{1}{{\sqrt {14} }},\cfrac{2}{{\sqrt {14} }},\cfrac{3}{{\sqrt {14} }} >$