# Angle Between Two Planes Calculator

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A free online calculator, showing all steps, to calculate the angle \( \alpha \) between two planes is presented.

## Formulas Used in the Calculator

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

\[ \large \color{red} {\cos \alpha = \dfrac{ \vec {n_1} \cdot \vec {n_2} }{| \vec {n_1} | \cdot | \vec {n_2} | } = \dfrac{a_1 \cdot a_2 + b_1 \cdot b_2 + c_1 \cdot c_2 }{| \vec {n_1} | \cdot | \vec {n_2} | } } \]
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
\[ \large \color{red} {\alpha = \arccos \left (\dfrac{ a_1 \cdot a_2 + b_1 \cdot b_2 + c_1 \cdot c_2 }{\sqrt {a_1^2 + b_1^2 + c_1^2 } \cdot \sqrt {a_2^2 + b_2^2 + c_2^2} } \right) } \]

## Use of the Calculator

Enter the coefficients \( a_1 \), \( b_1 \) and \( c_1 \) of plane (1) and coefficients \( a_2 \), \( b_2 \) and \( c_2 \) of plane (2) and press "Calculate". The outputs are the magnitudes \( | \vec {n_1} | \) and \( | \vec {n_2} | \), the dot product \( \vec {n_1} \cdot \vec {n_2} \) and angle \( \alpha \). You may also enter the number of decimal places required.

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