4.2. Two planes are given by their general equations:

$$
\begin{aligned}
& 5 x-4 y+z-1=0 \\
& 10 x+2 y+4 z-7=0
\end{aligned}
$$

Determine the magnitude of the dihedral angle between these planes.
Solution. The dihedral angle between planes is measured, as is known, by a linear angle, and the latter is equal to the angle between vectors perpendicular to the planes. The vector perpendicular to the first plane has coordinates $\vec{n}_{1}=(5,-4,1)$, to the second plane $\ddot{n}_{2}=(10,2,4)$. Let's use the formula for the cosine of the angle between two vectors

$$
\begin{aligned}
& \cos \varphi=\frac{\dot{n_{1}} \cdot \ddot{n}_{2}}{\left|\ddot{n}_{1}\right| \cdot\left|\vec{n}_{2}\right|}=\frac{5 \cdot 10+(-4) \cdot 2+1 \cdot 4}{\sqrt{5^{2}+(-4)^{2}+1^{2}} \cdot \sqrt{10^{2}+2^{2}+1^{2}}} \cong 0.69, \\
& \varphi=\arccos (0.69) \cong 52^{\circ} .
\end{aligned}
$$