Angles between lines and planes in space

An architect is designing the pitched roof of a sports hall. In the design, the main supporting structure follows the direction of the line $r: \frac{x-1}{2} = \frac{y+2}{1} = \frac{z}{2}$, while the floor of the hall lies on the horizontal plane $\pi: z = 0$. Calculate the angle formed by the support beam with the floor.

Calculate the angle formed by the line $r$ passing through the points $A(1, 0, 1)$ and $B(2, 1, 2)$ with the plane of general equation $\pi: x + y + 1 = 0$.

Calculate the acute angle formed by the line $r$, which passes through the points $A(1, 0, -2)$ and $B(3, -1, 1)$, with the plane $\pi$ of general equation $2x + y - 3z + 4 = 0$. Express the result in degrees, minutes and seconds.

Determine the value of the parameter $k$ so that the line $r: \frac{x-1}{k} = \frac{y}{2} = \frac{z+1}{-1}$ is perpendicular to the plane $\pi: 2x + 4y - 2z + 7 = 0$. What angle do they form in this case?

Given the line $r \equiv \begin{cases} x = 1 + \lambda \\ y = 2\lambda \\ z = -1 - \lambda \end{cases}$ and the plane $\pi \equiv x + ky + 2z - 1 = 0$, determine the value of the real parameter $k$ given that the angle formed by the line and the plane is $30^\circ$.

Calculate the sine of the angle formed by the line $r$, defined as the intersection of the planes $\pi_1: x - y = 0$ and $\pi_2: y - z = 0$, with the plane $\pi_3: x + y + 2z - 4 = 0$. (Hint: First find the direction vector of the line $r$ using the vector product of the normal vectors of $\pi_1$ and $\pi_2$).

Find the angle formed by the line $r$, defined as the intersection of the planes $\pi_1 \equiv x - y = 0$ and $\pi_2 \equiv 2y + z = 1$, with the plane $\alpha \equiv 3x - z = 0$.

A solar panel represented by the plane $\pi \equiv x + y + az + d = 0$ has been installed. It is known that the panel passes through the point $P(1, 2, 1)$ and that, to maximise its efficiency, it must form an angle of $60^\circ$ with the $OZ$ axis (the vertical axis).

Calculate the possible values of the parameter $a$ (assume $a > 0$).

Calculate the angle formed by the line $r \equiv \frac{x}{1} = \frac{y-1}{2} = \frac{z+1}{-1}$ with each of the coordinate planes. Do these three angles add up to $180^\circ$? Justify your answer using the concepts of direction and normal vectors.