Can a line in 3-space have all direction cosines $=\frac{1}{2}$

Question

I immediately found that it is impossible since the squares of the direction cosines have to add to 1 and $3 \times (\frac{1}{2})^2 \neq 1$. However, the textbook asks to "interpret geometrically", saying "Hint: Try holding a pencil so that it forms an angle of $\frac{\pi}{3}$ with each of the coordinate axes in space". How can I prove no such line exists without using any analytic geometry?