Finding angles between lines, between a line and a plane, and between two planes using direction vectors and normals, plus intersections of lines and planes and systems of three planes in 3D.
To find the intersection of a line and a plane, substitute the vector line equation into the plane equation and solve for the parameter λ.
The angle ϕ between a line and a plane is measured as the complement of the angle between the line’s direction vector d and the plane’s normal vector n.
Since the normal forms an angle of 90° with the plane, we can construct a right angled triangle using the intersection of the normal, the line, and the plane:
If θ is the angle between d and n, then ϕ=180°−90∘−θ=90°−θ
In practice, you can compute θ using cosθ=∣d∣∣n∣d⋅n.
The intersection of 2 planes - if they are not parallel - is a line. We can find the line intersection by solving a system of equations.
The angle θ between two planes can be found by computing the angle between the two plane normals:
The diagram below illustrates why this works. Note that the formula above gives the angle between the "heads" of the normal vectors, which can either be acute or obtuse. Just like the angle between lines, if the question asks for an acute angle and cos−1 returns an obtuse angle, just subtract it from 180∘ to get the acute angle.
Systems of 3 equations with 3 unknowns, for example
can be solved with a calculator, or by using substitution.
For the system above, equation (2) is a convenient place to start, because it contains just −z, so it is easy to rearrange for z:
Now substitute this expression for z into equations (1) and (3).
Using equation (3):
We know have a system of 2 equations with 2 unknowns. Eliminate y by taking
Now substitute x=1 into 13x+8y=29:
Finally, substitute x=1 and y=2 into z=5x+2y−6:
So the solution is x=1,y=2,z=3.
A system of 3 equations with 3 unknowns can have
no solutions
a unique solution
infinitely many solutions
Consider the system of equations
Add equations (1) and (2):
Now add equations (2) and (3):
Substitute y=2z:
If 2a−1=0, then we can solve for
and then find y and z. This represents a unique solution.
But if 2a−1=0, ie a=21, then we have 0z=1, to which there are no solutions.
When three planes are considered together, their intersection in 3D space can take several forms:
A plane if all three planes coincide.
A single point if the three planes intersect uniquely, meaning their normal vectors are not all parallel or in some degenerate arrangement, and the system of equations has exactly one solution.
A line if two planes intersect in a line and the third plane also contains that line (or if each pair of planes meets along the same line).
No intersection if the system of equations is inconsistent (e.g., the planes are arranged in parallel or partially parallel ways that do not share a common point).
Nice work completing Intersections with planes, here's a quick recap of what we covered:
Exercises checked off
Finding angles between lines, between a line and a plane, and between two planes using direction vectors and normals, plus intersections of lines and planes and systems of three planes in 3D.
To find the intersection of a line and a plane, substitute the vector line equation into the plane equation and solve for the parameter λ.
The angle ϕ between a line and a plane is measured as the complement of the angle between the line’s direction vector d and the plane’s normal vector n.
Since the normal forms an angle of 90° with the plane, we can construct a right angled triangle using the intersection of the normal, the line, and the plane:
If θ is the angle between d and n, then ϕ=180°−90∘−θ=90°−θ
In practice, you can compute θ using cosθ=∣d∣∣n∣d⋅n.
The intersection of 2 planes - if they are not parallel - is a line. We can find the line intersection by solving a system of equations.
The angle θ between two planes can be found by computing the angle between the two plane normals:
The diagram below illustrates why this works. Note that the formula above gives the angle between the "heads" of the normal vectors, which can either be acute or obtuse. Just like the angle between lines, if the question asks for an acute angle and cos−1 returns an obtuse angle, just subtract it from 180∘ to get the acute angle.
Systems of 3 equations with 3 unknowns, for example
can be solved with a calculator, or by using substitution.
For the system above, equation (2) is a convenient place to start, because it contains just −z, so it is easy to rearrange for z:
Now substitute this expression for z into equations (1) and (3).
Using equation (3):
We know have a system of 2 equations with 2 unknowns. Eliminate y by taking
Now substitute x=1 into 13x+8y=29:
Finally, substitute x=1 and y=2 into z=5x+2y−6:
So the solution is x=1,y=2,z=3.
A system of 3 equations with 3 unknowns can have
no solutions
a unique solution
infinitely many solutions
Consider the system of equations
Add equations (1) and (2):
Now add equations (2) and (3):
Substitute y=2z:
If 2a−1=0, then we can solve for
and then find y and z. This represents a unique solution.
But if 2a−1=0, ie a=21, then we have 0z=1, to which there are no solutions.
When three planes are considered together, their intersection in 3D space can take several forms:
A plane if all three planes coincide.
A single point if the three planes intersect uniquely, meaning their normal vectors are not all parallel or in some degenerate arrangement, and the system of equations has exactly one solution.
A line if two planes intersect in a line and the third plane also contains that line (or if each pair of planes meets along the same line).
No intersection if the system of equations is inconsistent (e.g., the planes are arranged in parallel or partially parallel ways that do not share a common point).
Nice work completing Intersections with planes, here's a quick recap of what we covered:
Exercises checked off