Exercises: 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.

The flight path of a surveillance drone is defined by the line $r: \begin{cases} x = 1 + \lambda \\ y = 2 - \lambda \\ z = 3\lambda \end{cases}$. The drone must pass through a reinforced glass panel located in the plane $\pi: 2x + y - z + 4 = 0$. Determine the angle of incidence (the angle formed by the path with the plane) at which the drone will hit the glass.

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 $\pi: x + y + 1 = 0$.

Given the line $s: \frac{x}{1} = \frac{y-3}{-1} = \frac{z+2}{2}$ and the plane $\pi: 3x - 2y + z - 5 = 0$, check whether the line and the plane are parallel or, on the contrary, intersect at a specific angle. If they intersect, calculate this angle.

A laser beam leaves the origin $O(0, 0, 0)$ in the direction of the vector $\vec{v} = (1, 1, 1)$. Consider a flat wall whose equation is $\pi: x - 2y + z = 0$. What angle does the light beam form with the surface of the wall?

Find the angle between the line $r: \begin{cases} x + y = 2 \\ z = 1 \end{cases}$ and the coordinate plane $OXY$ ($z = 0$). Briefly explain the geometric meaning of the result obtained.

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?

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 $\pi_2$).

Marcos is faced with the following exam problem: "Calculate the angle $\alpha$ between the line $r: \frac{x-2}{1} = \frac{y+1}{0} = \frac{z}{0}$ and the plane $\pi: x + \sqrt{3}y - 7 = 0$". In his working, he writes:

"I extract the direction vector of the line $\vec v = (1, 0, 0)$ and the normal vector of the plane $\vec n = (1, \sqrt{3}, 0)$. I apply the dot product formula:

Therefore, $\alpha = \arccos(1/2) = 60^\circ$".

1. a) Point out the conceptual mistake Marcos made and explain geometrically why the appropriate trigonometric ratio between a line and a plane is the sine and not the cosine.
2. b) Correct the procedure and find the actual value of the angle $\alpha$.
3. c) If the dot product between $\vec v$ and $\vec n$ had been equal to $0$, what relative position would the line and the plane have? What angle would they form?

Consider the line $r$ passing through the origin of coordinates $O(0, 0, 0)$ and the point $A(1, 1, 8)$, and the horizontal ground plane $\pi: z = 0$.

1. a) Estimate: Before performing any trigonometric calculation, analyze the components of the direction vector $\vec{OA} = (1, 1, 8)$. If a point ascends $8$ units vertically while advancing only $\sqrt{1^2+1^2} = \sqrt{2} \approx 1,41$ units in the horizontal plane, do you reasonably expect the angle $\alpha$ of the line with the ground to be closer to $15^\circ$, $45^\circ$, or $80^\circ$? Qualitatively justify your prediction.
2. b) Calculate: Analytically obtain the exact value of $\sin\alpha$ using the corresponding vectors and calculate the value of $\alpha$ rounded to the nearest minute of a degree.
3. c) Check: Compare the result with your initial estimate. Explain what would happen to the angle $\alpha$ if the $z$-coordinate of point $A$ increased indefinitely while keeping its $x$- and $y$-coordinates constant.

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.

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 $30^\circ$.

An architectural firm designs a skylight on a sloping roof represented by the plane $\pi \equiv 3x + 4y + z - 12 = 0$. A ray of light strikes along the direction of the line $s \equiv \frac{x-1}{2} = \frac{y}{-1} = \frac{z+2}{2}$.

Calculate the angle $\alpha$ that the ray of light forms with the plane of the roof and the angle of incidence $\beta$ that it forms with the normal to the plane.

Determine the Cartesian equation of the plane $\pi$ that contains the line $r \equiv \frac{x-1}{2} = \frac{y+2}{3} = \frac{z}{1}$ and is perpendicular to the plane $\sigma \equiv x - y + z + 3 = 0$. Once found, check what angle the line $r$ forms with that plane $\pi$. Was this result expected? Justify your answer.

Consider the cube of edge length 1 unit with a vertex at the origin $(0,0,0)$ and edges along the positive coordinate axes. Let $d$ be the line containing the main diagonal of the cube (joining $(0,0,0)$ to $(1,1,1)$).

Find the angle formed by this diagonal $d$ with the face of the cube lying in the plane $XY$.

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.

It is planned to design a straight loading ramp $r$ starting from the anchor point $P(1, 0, 2)$. Due to architectural constraints of the industrial warehouse, the trajectory of the ramp must be entirely contained in the vertical plane $\pi_1: x = 1$ and, additionally, ergonomic protocols require it to ascend forming an angle of exactly $30^\circ$ with the horizontal floor $\pi_2: z = 0$.

1. a) From the condition of lying in the plane $\pi_1: x = 1$, deduce which component of the direction vector $\vec v = (v_1, v_2, v_3)$ is necessarily zero.
2. b) Set up the equation for the angle between a line and a plane with $\pi_2: z = 0$ to find the components of the possible direction vectors $\vec v$ (assuming, to fix the scale, that the vertical component is $v_3 = 1$).
3. c) Write the continuous (symmetric) equations of the two distinct lines that solve the problem and explain geometrically why exactly two solutions appear that are symmetric with respect to the vertical line through $P$.

In an agro-energy irrigation farm in the province of Jaén, a solar panel is installed, oriented to maximize summer collection. In the fixed coordinate system (with elevations in meters and the ground located at the horizontal plane $z = 0$), the photovoltaic panel rests on the plane $\pi: 2x - y + 2z - 6 = 0$.

To reinforce the structure against the wind, a straight steel bar $r$ is anchored, connecting the ground point $A(1, 0, 0)$ to the upper vertex of the structure $B(3, 2, 4)$.

1. a) Determine the symmetric equations of the line $r$ that models the bar and the components of a normal vector $\vec n$ to the plane $\pi$ of the panel.
2. b) Calculate the angle $\alpha \in [0^\circ, 90^\circ]$ formed by the bar and the flat surface of the solar panel.
3. c) The project's technical specifications warn that, to prevent tearing stress on the panel anchors, the reinforcement bar must meet the surface at an angle strictly greater than $20^\circ$. Determine, with justification, whether the executed installation complies with this technical requirement.

A laser beam is emitted from a satellite along the path of the line $r$, which passes through the point $A(1, 0, 2)$ and has direction vector $\vec{v}(k, 1, 1)$. The goal is for the beam to hit a receiver station located on a plane $\pi$ with equation $2x - y + z = 5$. Determine the values of the real parameter $k$ so that the angle formed by the laser path with the receiver plane is exactly $30^\circ$.

In the design of a state-of-the-art auditorium, a support beam is planned along the direction of the line $s$, defined by the intersection of the planes:

$s \equiv \begin{cases} x + y - z = 0 \\ 2x - y + 3z = 1 \end{cases}$

It is required that this beam forms an angle $\alpha$ with the plane of the main facade $\pi \equiv x + ay + z = 3$. Knowing that $\sin(\alpha) = \frac{\sqrt{6}}{6}$, calculate the value of the coefficient $a$ of the facade.

An industrial robotic arm in an assembly plant must move along a straight guide $r$ that passes through the origin $O(0,0,0)$ and the point $P(1, -1, 0)$. The arm must operate on an inclined metal plate that coincides with the plane $\pi$ containing the points $A(2, 0, 0)$, $B(0, 2, 0)$ and $C(0, 0, 2)$. Calculate the angle that the robot's path forms with the workpiece plate.

A charged particle moves in an accelerator along the line $r$ passing through the point $M(2, 3, -1)$ and parallel to the line $s \equiv \frac{x-1}{2} = \frac{y+2}{1} = \frac{z}{2}$. Upon entering a region with a magnetic field, the particle crosses a plane $\pi$ perpendicular to the line $t \equiv (x,y,z) = (1, 0, 0) + \lambda(1, -2, 2)$. Determine the entry angle of the particle into this magnetic field (the angle between $r$ and $\pi$).

Consider a pyramid whose base is the plane $XY$ ($z=0$) and whose main slant edge lies on the line $r$ joining the apex $V(2, 2, 5)$ to the base point $P(0, 0, 0)$.

1. a) Find the angle formed by this edge with the plane of the base.
2. b) If we wanted to tilt the plane of the base so that the angle with the edge was $90^\circ$, what should the equation of the new base plane be if it must still pass through the origin?

Find the equation of the line $r$ passing through the point $Q(1, 1, 1)$, parallel to the plane $\alpha \equiv x - y + z = 2$ and forming an angle of $45^\circ$ with the plane $\beta \equiv x + y = 0$. Is this line unique? Justify your answer using the concept of direction vectors.

A solar panel is mounted on a metal structure. The panel is contained in the plane $\pi \equiv x + y + z = 1$. The sun rays at noon follow the direction of the line $r$ passing through the point $S(10, 10, 10)$ and perpendicular to the terrain's slope plane $\sigma \equiv 2x - y = 0$.

1. a) Calculate the angle of incidence of the sun rays on the panel.
2. b) If in order to obtain maximum efficiency the angle must be $90^\circ$, determine a normal vector for a new panel that satisfies this condition.

Calculate the angle formed by the line $r$ with the plane $\pi$, given that $r$ is the line containing the median from vertex $A$ in the triangle with vertices $A(1, 2, 1)$, $B(3, 0, 1)$ and $C(1, 4, 3)$, and that the plane $\pi$ is the plane containing the points $B$, $C$ and the origin $O(0, 0, 0)$.

Given the line $r: \frac{x-3}{2} = \frac{y+1}{-1} = \frac{z-4}{2}$ and the horizontal $XY$ plane with general equation $\pi: z = 0$.

To find the angle $\alpha$ that the line $r$ forms with said plane, two 2nd-year Bachillerato students propose alternative approaches:

• Strategy A (Line-plane angle): Use the normal vector of the plane $\vec n = (0, 0, 1)$ and the direction vector $\vec v = (2, -1, 2)$ via the direct formula $\sin\alpha = \frac{|\vec v \cdot \vec n|}{|\vec v|\cdot|\vec n|}$.

• Strategy B (Complementary angle with the $OZ$ axis): Treat the vertical $OZ$ axis as a line with direction vector $\vec u = (0, 0, 1)$, calculate the angle between lines $\beta$ via $\cos\beta = \frac{|\vec v \cdot \vec u|}{|\vec v|\cdot|\vec u|}$, and finally obtain $\alpha = 90^\circ - \beta$.

1. a) Fully work out both methods numerically and verify that they yield exactly the same measure for $\alpha$.
2. b) Analytically prove that the formula in Strategy A can be deduced from Strategy B by applying the trigonometric identity $\cos(90^\circ - \alpha) = \sin\alpha$, explaining the role of the complement in the metric definition.
3. c) Critically evaluate both procedures: if the plane in the problem were not horizontal but any oblique plane (for example, $\pi': 2x - 3y + z = 5$), what insurmountable disadvantage or operational overhead would Strategy B have compared to Strategy A? Justify which general method is preferable.

Consider the line $s$ defined by the intersection of planes:

and the line $r$ given in continuous (symmetric) form:

The pencil of planes intersecting along line $s$ is defined by the equation $\pi_\lambda: (x - z) + \lambda(y - 3) = 0$ with $\lambda \in \mathbb{R}$, together with the complementary plane $\pi_\infty: y - 3 = 0$.

1. a) Find the values of $\lambda$ for which the angle $\alpha$ formed by the line $r$ and the plane $\pi_\lambda$ is exactly $45^\circ$. Write the general equation of each of the planes obtained.
2. b) Analytically check whether the complementary plane $\pi_\infty: y - 3 = 0$ also forms an angle of $45^\circ$ with $r$.
3. c) Reason geometrically whether there is any plane in this pencil that can be strictly perpendicular to the line $r$. To do so, analyze the necessary condition that the direction vectors of both lines, $\vec v_s$ and $\vec v_r$, would have to satisfy, without needing to solve complex systems of equations.