Mathematics

The key is to use the constraint that $\boldsymbol{a}$ is a unit vector with a fixed angle to the Z-axis, then apply the condition that $\boldsymbol{a} + \hat{\boldsymbol{i}} + \hat{\boldsymbol{j}}$ is also unit. Write $\boldsymbol{a} = (x, y, z)$ where $x^2 + y^2 + z^2 = 1$ and $z = \cos(\pi/4) = 1/\sqrt{2}$. The second constraint—that the sum has magnitude 1—gives you an equation in $x$ and $y$; symmetry and the geometry of the problem will pin down the unique solution.

The key is to use the constraint that $\boldsymbol{a}$ is a unit vector with a fixed angle to the Z-axis, then apply the condition that $\boldsymbol{a} + \hat{\boldsymbol{i}} + \hat{\boldsymbol{j}}$ is also unit. Write $\boldsymbol{a} = (x, y, z)$ where $x^2 + y^2 + z^2 = 1$ and $z = \cos(\pi/4) = 1/\sqrt{2}$. The second constraint—that the sum has magnitude 1—gives you an equation in $x$ and $y$; symmetry and the geometry of the problem will pin down the unique solution.