If $\theta$ is the acute angle between the diagonals of a cube, then which one of the following is correct?

Consider a cube with side length $a$ and one vertex at the origin $(0,0,0)$.Two main diagonals are:From $(0,0,0)$ to $(a,a,a)$. Vector $\overrightarrow{d_1}=a\hat{i}+a\hat{j}+a\hat{k}$.From $(a,0,0)$ to $(0,a,a)$. Vector $\overrightarrow{d_2}=-a\hat{i}+a\hat{j}+a\hat{k}$.The angle $\theta$ between them is given by:
$$\cos \theta =\frac{|\overrightarrow{d_1}\cdot \overrightarrow{d_2}|}{|\overrightarrow{d_1}||\overrightarrow{d_2}|}$$
$$\overrightarrow{d_1}\cdot \overrightarrow{d_2}=(a)(-a)+(a)(a)+(a)(a)=-a^2+a^2+a^2=a^2$$
$$|\overrightarrow{d_1}|=\sqrt{a^2+a^2+a^2}=a\sqrt{3}$$
$$|\overrightarrow{d_2}|=\sqrt{(-a{)}^2+a^2+a^2}=a\sqrt{3}$$
$$\cos \theta =\frac{a^2}{(a\sqrt{3})(a\sqrt{3})}=\frac{a^2}{3a^2}=\frac{1}{3}$$
$$3\cos \theta =1$$