Solution

Consider $$\vec {p}=\left (\hat {i}+\hat {j}+\hat {k}\right )$$ and $$\vec {q}=\left (2\hat {i}+2\hat j+2\hat k\right )$$
The directions cosines of $$\vec {p}$$ are given by,
$$l=\dfrac {1}{\sqrt {1^2+1^2+1^2}}=\dfrac {1}{\sqrt {3}}, m=\dfrac {1}{\sqrt {1^2+1^2+1^2}}=\dfrac {1}{\sqrt {3}}$$, and $$n=\dfrac {1}{\sqrt {1^2+1^2+1^2}}=\dfrac {1}{\sqrt {3}}$$.
The direction cosines of $$\vec {q}$$ are given by
$$l=\dfrac {2}{\sqrt {2^2+2^2+2^2}}=\dfrac {2}{2\sqrt 3}=\dfrac {1}{\sqrt {3}}, m=\dfrac {2}{\sqrt {2^2+2^2+2^2}}=\dfrac {2}{2\sqrt 3}=\dfrac {1}{\sqrt {3}}$$, and $$n=\dfrac {2}{\sqrt {2^2+2^2+2^2}}=\dfrac {2}{2\sqrt 3}=\dfrac {1}{\sqrt {3}}$$.
The direction cosines of $$\vec {p}$$ and $$\vec {q}$$ are the same. Hence, the two vectors have the same direction.