Section 1.6 Equations of Planes

1.6 Equations of Planes

In this section, we introduce equations of planes in a 3-dimensional or higher space. Recall that given two distinct points [latex]P=(x_{1},y_{1},z_{1})[/latex] and [latex]Q=(x_{2},y_{2},z_{2})[/latex], we can define a line passing those two points. This is saying two points decide a line. Three points, [latex]P=(x_{1},y_{1},z_{1})[/latex], [latex]Q=(x_{2},y_{2},z_{2})[/latex] and [latex]R=(x_{3},y_{3},z_{3})[/latex], give a plane in the space. We know that the direction of two points give the direction of the line passing those two points. We now have three points, we have three directions and three lines. Which direction we should use for the plane passing those three points? There is a unique direction, normal vector, such that it is perpendicular to all three lines. Not only that, any two points on this plane give a direction and it must perpendicular to this normal vector as well. This is saying that a plane is actually decided by a normal vector, [latex]\overrightarrow{n}[/latex] and a point, [latex]P=(x_{1},y_{1},z_{1})[/latex] on the plane. 

Definition: 

(a) Given a point [latex]P=(x_{1},y_{1},z_{1})[/latex] and a vector [latex]\overrightarrow{n}=\lt a,b,c\gt[/latex], the set of all points [latex]Q=(x,y,z)[/latex] satisfying [latex]\overrightarrow{PQ}\cdot\overrightarrow{n}=0[/latex] form a plane. The equation

\[\overrightarrow{n}\cdot\overrightarrow{PQ} =\lt a,b,c\gt \cdot \lt x-x_{1},y-y_{1},z-z_{1}\gt =0 \]

is called the vector equation of a plane. 

(b) The vector equation of the plane can be simplified as 

\[ a(x-x_{1})+b(y-y_{1})+c(z-z_{1})=0 \]

 which is called scalar equation of a plane. 

(c) We can simplified more to obtain 

\begin{align*} ax+by+cz+(-ax_{1}-by_{1}-cz_{1}) & =0\\ ax+by+cz+d & =0 \end{align*}

where [latex]d=-ax_{1}-by_{1}-cz_{1}[/latex] and the equation is called the general form of the equation of a plane.

Example 1: Let [latex]P=(1,-2,3)[/latex] and [latex]\overrightarrow{n}=\lt 2,1,0\gt[/latex]. Find the vector equation, scalar equation, and general form of the equation of a plane that is perpendicular to the vector [latex]\overrightarrow{n}[/latex] and passing the point [latex]P[/latex].

Exercise 1: Let [latex]P=(-1,2,-3)[/latex] and [latex]\overrightarrow{n}=\lt 0,3,1\gt[/latex]. Find the vector equation, scalar equation, and general form of the equation of a plane that is perpendicular to the vector [latex]\overrightarrow{n}[/latex] and passing the point [latex]P[/latex].

Example 2: Let [latex]P=(1,-2,3)[/latex], [latex]Q=(2,1,0)[/latex] and [latex]R=(0,1,2)[/latex]. Find the general equation of the plane passing through the given three points. 

Exercise 2: Let [latex]P=(-1,2,-3)[/latex], [latex]Q=(0,3,1)[/latex] and [latex]R=(1,0,-1)[/latex]. Find the general equation of the plane passing through the given three points. 

Example 3: Let [latex]P=(1,-2,3)[/latex] and [latex]L:\frac{x-2}{1}=\frac{y}{2}=\frac{z-1}{-2}[/latex]. Find the general equation of the plane passing through [latex]P[/latex] and is perpendicular to [latex]L[/latex]. 

Exercise 3: Let [latex]P=(-1,2,0)[/latex] and [latex]L:\frac{x}{3}=\frac{y-1}{1}=\frac{z-1}{-2}[/latex]. Find the general equation of the plane passing through [latex]P[/latex] and is perpendicular to [latex]L[/latex]. 

Example 4: Let [latex]P=(1,2,-3)[/latex] and [latex]L:\frac{x-2}{1}=\frac{y}{2}=\frac{z-1}{-2}[/latex]. Find the general equation of the plane passing through [latex]P[/latex] and containing [latex]L[/latex]. 

Exercise 4: Let [latex]P=(-1,2,0)[/latex] and [latex]L:\frac{x}{3}=\frac{y-1}{1}=\frac{z-1}{-2}[/latex]. Find the general equation of the plane passing through [latex]P[/latex] and containing [latex]L[/latex]. 

Definition: Two planes are parallel to each others if their normal vectors are parallel to each others. 

Example 5: Find the general equation of the plane that passing the point [latex](2,3,0)[/latex] and is parallel to the plane [latex]z=3x+4y[/latex].

Exercise 5: Find the general equation of the plane that passing the point [latex](0,1,-2)[/latex] and is parallel to the plane [latex]z=-2x+y[/latex].

Example 6: Find the general equation of the plane that containing [latex]L:\frac{x-3}{2}=\frac{y}{1}=\frac{z-2}{-4}[/latex] and is parallel to the plane [latex]z+4y=0[/latex].

Exercise 6: Find the general equation of the plane that containing [latex]L:\frac{x+1}{-3}=\frac{y+2}{-6}=\frac{z-2}{1}[/latex] and is parallel to the plane [latex]-2x+y=0[/latex].

Fact: Given any two planes, they are either parallel or they intersect with an angle which is decided by the angle of the normal vectors of two planes.

Example 7: Decide if given two planes, [latex]x-z=1[/latex] and [latex]2x-y+2z=2[/latex] are parallel or perpendicular or neither. If neither, find the angle of the intersection. 

Exercise 7: Decide if given two planes, [latex]2x+y=1[/latex] and [latex]2x-y+z=1[/latex] are parallel or perpendicular or neither. If neither, find the angle of the intersection. 

Example 8: Where does the line through [latex]P=(2,1,0)[/latex] and [latex]Q=(0,-2,1)[/latex] intersect the plane [latex]2x-y+z=2[/latex]?

Example 9: Find the angle of the intersection of given two planes and find the parametric equation of the intersection line of the two planes. [latex]x-2y+z=0[/latex] and [latex]2x-z=0[/latex].

Group work:

1. Where does the line through [latex]P=(2,0,-1)[/latex] and [latex]Q=(1,1,0)[/latex] intersect the plane [latex]y+z=1[/latex]?

2. Find the angle of the intersection of given two planes and find the parametric equation of the intersection line of the two planes. [latex]2x-y=0[/latex] and [latex]x+2z=0[/latex].

3. Sketch the plane [latex]2y+3z=6[/latex].

4. Sketch the plane [latex]2x-y+3z=6[/latex].

5. Find the plane passing through [latex]P=(1,1,0)[/latex], [latex]Q=(0,1,1)[/latex] and [latex]R=(1,0,1)[/latex].

6. Consider line [latex]L[/latex] of symmetric equations [latex]x-2=-y=\frac{z}{2}[/latex] and point [latex]A(1,1,1)[/latex]. Find the plane passing [latex]A[/latex] and containing [latex]L[/latex]. 

7. Consider line [latex]L[/latex] of symmetric equations [latex]x-2=-y=\frac{z}{2}[/latex] and point [latex]A(1,1,1)[/latex]. Find the plane passing [latex]A[/latex] and is perpendicular to [latex]L[/latex].