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We already know how to find the intersection and the angle between two lines. We can apply a similar idea to the intersection and angle between lines and planes or between two planes.
Discussion
A line which is not parallel to a plane will intersect it at a single point whose coordinates satisfy the equations of both the line and the plane.
Let a line is given by r=⎝⎛324⎠⎞+λ⎝⎛−112⎠⎞ and a plane by Π:x−y+2z=1.
How would we find the point of intersection between r and Π? Think about what would need to be true about this point. Can you set up an equation that would give the coordinates of the point as a solution?
Any point on the line can be written as
For this point to also lie on the plane x−y+2z=1, its coordinates must satisfy the plane equation.
The general method is:
Write the general point on the line: (3−λ,2+λ,4+2λ).
Substitute these into the plane equation and solve for λ:
Once you find λ, substitute it back into the line equation to get the intersection point.
Intersection of line and plane
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.
Example
Suppose we have the line:
and the plane:
Substituting the line into the plane gives:
Expanding:
Substituting this back into the line gives the intersection point:
Exercise
Find the point of intersection of the line r=⎝⎛32−1⎠⎞+λ⎝⎛−121⎠⎞ and the plane 3x−y+z=2.
Select the correct option
We don't actually need to find the point of intersection to find the angle between a line and a plane. Since the orientation of a line is defined by its direction vector and the orientation of a plane is defined by its normal vector (where "orientation" here means "direction in space"), finding the angle between these two will allow us to find the angle between the line and the plane. However, since the normal vector of a plane is perpendicular to that plane, we can't just give the angle between the normal vector and the line's direction vector as our answer, since this represents a different quantity. Instead, we use trigonometry to solve for the actual angle once we know an angle from this formula.
Angle between line and plane
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.
Note: Here, as with all other spatial models in this section, we encourage you to close your eyes and visualize why this formula works. Being able to picture these ideas without a drawing in front of you will allow you to build intuition, understanding the formulas instead of just memorizing them so that when you're asked questions relating to these formulas, you can better grasp how to apply them in that specific case.
Example
Consider the line r=⎝⎛000⎠⎞+λ⎝⎛231⎠⎞ and the plane with normal n=⎝⎛314⎠⎞. The direction vector is d=⎝⎛231⎠⎞.
The angle θ between d and n satisfies
So θ=47.05°, thus ϕ=90°−47.05°=43.0°
Checkpoint
A line l and plane Π have direction vector d and normal vector n, respectively. It is given that ∣d∣=2,∣n∣=5, and d⋅n=5√2.
Find the angle between l and Π.
Select the correct option
Exercise
Find the angle between the line r=⎝⎛2−21⎠⎞+λ⎝⎛03−5⎠⎞ and the plane Π:x−3y+2z=6.
Select the correct option
We can also find the intersection between two planes. (Since finding this line will constitute a system of three equations in two unknowns, see the example to understand how we would go about finding this line.)
Intersection between 2 planes
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.
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Example
Consider the system of planes:
We have two equations in three unknowns. First, we need to eliminate one of the variables. Adding two times the second equation to the first equation will allow us to eliminate y (we can eliminate any of the three, but this is the simplest in this case):
Let z=λ. (This is necessary since we have three equations in two unknowns, which means it's impossible to find a single solution to this system, but that's okay: we're looking for a line that defines infinite solutions. Since our final line will include the parameter λ (or t, or μ, or whatever you'd like to call it), it's fair game to replace one of the variables with this parameter. This will just mean that in our final solution, the position vector will have z=0 and the direction vector will have z=1λ.) Then
Now we substitute x=2−λ AND z=λ into the equation 2x−y+z=3:
This gives the vector
which defines a line. Finally, we rewrite this in the more conventional way by separating the direction and position vectors. Thus the line of intersection between the two planes is:
The specific values will of course change for any two planes, but these steps work in every case where the system has solutions.
Exercise
Find the equation of the line of intersection between the planes 4x−y+3z=2 and x+2y−z=8.
Select the correct option
Discussion
Consider two planes, Π1 and Π2, and their normal vectors n1 and n2.
How would you find the angle between their normal vectors?
Is this the same angle as the one between the planes themselves, or different? Why?
To measure how “far apart” the two normal‐lines n1 and n2 point, we use their dot product. Recall that for any non‐zero vectors u and v,
where θ is the angle between them. Applying this to n1 and n2 gives
so
measures the angle between the two normals.
Why does this coincide with the angle between the planes? Each normal is perpendicular to its plane, so turning one normal toward the other by θ “opens up” the same dihedral angle between the two planes. In fact, if you ever get an obtuse θ>2π, the acute angle between the planes is
but one often packages this by taking absolute value in the dot‐product formula and writing
so that 0≤ϕ≤2π. In other words, the (acute) angle between the planes is the same as the (acute) angle between their normals.
Angle between 2 planes
The angle θ between two planes can be found by computing the angle between the two plane normals,
Example
Find the angle between the planes
The normals are n1=⎝⎛21−1⎠⎞ and n2=⎝⎛2−11⎠⎞, both with magnitude √6. So
So
Checkpoint
Two planes have normal vectors n1 and n2 such that
Find the angle between the planes.
Select the correct option
Exercise
Find the acute angle between planes with equations 7x+2y−z=9 and −3x+5y+4z=6.
Select the correct option
There are more possible configurations for three planes than for two. Because of this, when asked to find an intersection between three planes, you need to first determine how many solutions (zero, one, or infinite) the system has, and then understand what this tells you in the context of the problem. With three planes, there is more than one "way" for their system to have zero or infinite solutions.
As you look at the diagrams below, try to imagine what a solution to their systems represents.
Intersection of 3 planes
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.
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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).
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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).
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Exercise
Three planes are given by the equations
Find the equation of the line where the planes intersect.
Select the correct option
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