Angles Between Lines and Planes in Space
A solar panel on a rooftop in Mérida must receive the sun's rays at a precise tilt, neither too glancing nor too vertical. The same goes for an antenna, a roof, or a robot's arm on an assembly line. That angle, the one a straight path makes with a flat surface, is always calculated with the same idea: compare the direction vector of the line with the normal vector of the plane.
The Angle Between a Line and a Plane
An architect is designing the pitched roof of a sports hall. In the design, the main supporting structure follows the direction of the line…
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 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…
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.
Special Cases: Perpendicular or Parallel
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 $\pi: 2x + 4y - 2z + 7 = 0$.…
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…
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 $30^\circ$.
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…
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 $30^\circ$.
A Line as the Intersection of Two Planes
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…
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…
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 $\alpha \equiv 3x - z = 0$.
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:…
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.
How to Get the Plane You Need
Many problems don't give you the plane already written out: they give you three points, or a line and a condition. Before computing the angle you have to build 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…
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.
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…
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.
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…
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 $O(0, 0, 0)$.
Angle with the Coordinate Planes and with an Axis
Back to the start: to find the tilt at which the rays hit a panel, you compute the direction vector of the path, the normal of the panel's plane, and apply $\sin\alpha=\dfrac{|\vec v\cdot\vec n|}{|\vec v||\vec n|}$.
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…
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…
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.
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 $P(0, 0, 0)$. a)…
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 $P(0, 0, 0)$.
- a) Find the angle formed by this edge with the plane of the base.
- 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?