Plane (1) and plane (2) have the following equations \( \quad a_1 x + b_1 y + c_1 + d_1 = 0\) and \( \quad a_2 x + b_2 y + c_2 + d_2 = 0\) respectively.
The vectors \( \vec {n_1} \) and \( \vec {n_2} \) normal to the planes (1) and (2) defined by their equation above are given by their components as:
\( \vec {n_1} \; = \; \lt a_1 , b_1 , c_1 \gt \)
\( \vec {n_2} \; = \; \lt a_2 , b_2 , c_2 \gt \)
Angle \( \alpha \) between the two planes is equal to the angle between vectors \( \vec {n_1} \) and \( \vec {n_2} \) and its cosine is given by
\[ \large \color{red} {\cos \alpha = \dfrac{ \vec {n_1} \cdot \vec {n_2} }{| \vec {n_1} | \cdot | \vec {n_2} | } = \dfrac{a_1 \cdot a_2 + b_1 \cdot b_1 + c_1 \cdot c_2 }{| \vec {n_1} | \cdot | \vec {n_2} | } } \] The magnitudes \( | \vec {n_1} | \) and \( | \vec {n_2} | \) are given by
\( | \vec {n_1} | = \sqrt {a_1^2 + b_1^2 + c_1^2 } \)
\( | \vec {n_2} | = \sqrt {a_2^2 + b_2^2 + c_2^2 } \)
Use the inverse cosine function to express angle \( \alpha \) made by the two vectors as \[ \large \color{red} {\alpha = \arccos \left (\dfrac{ a_1 \cdot a_2 + b_1 \cdot b_1 + c_1 \cdot c_2 }{\sqrt {a_1^2 + b_1^2 + c_1^2 } \cdot \sqrt {a_2^2 + b_2^2 + c_2^2} } \right) } \]
