How to know if plane is perpendicular to another plane?

In summary, perpendicular planes are two planes that intersect at a right angle, and their equations can be determined to be perpendicular by finding the dot product of their normal vectors. The slopes of perpendicular planes are negative reciprocals of each other, and two planes cannot be perpendicular if they are parallel. To visualize perpendicular planes, imagine two intersecting sheets of paper forming an "L" shape.
  • #1
nicole1
1
0
The question that I'm trying to answer states "Make a vector equation of a plane that is perpendicular to the z axis." How do i ensure its perpendicular? How do i start this equation?

Another question similar to this that i am also struggling states "What is the vector equation of a 2D line that is perpendicular to this line: 2x + 3y = 6 though the lines y intercept?"

Any help for both of these questions would be greatly appreciated!
 
Mathematics news on Phys.org
  • #2
nicole said:
The question that I'm trying to answer states "Make a vector equation of a plane that is perpendicular to the z axis." How do i ensure its perpendicular? How do i start this equation?

Another question similar to this that i am also struggling states "What is the vector equation of a 2D line that is perpendicular to this line: 2x + 3y = 6 though the lines y intercept?"

Any help for both of these questions would be greatly appreciated!

Find a NORMAL VECTOR to the Planes. If they are orthogonal, you're done.

Rather the same deal for the lines. Compare the slopes.
 
  • #3
Any plane can be written in the form Ax+ By+ Cz= D for constants A, B, C, and D. And then <A, B, C> is a vector normal (perpendicular) to that plane.

The z-axis has "unit direction vector" <0, 0, 1>. A plane perpendicular to it must have that as "normal vector" and so the equation of the plane can be taken as 0x+ 0y+ 1z= z= C for some constant C.

The line 2x+ 3y= 6 has y-intercept (0, 2) (the y-intercept of a line is the point where it crosses the y-axis so x= 0). Further, if we were to "solve" that equation for y, 3y= 6- 2x, so y= 2- (2/3)x. That shows, again, that the y-intercept is 2 and that the slope is -2/3. A line perpendicular to the given line must have slope 3/2. Any (non-vertical) line can be written in the form y= mx+ b where m is the slope and b is the y-intercept. A line with slope 3/2 and y-intercept 2 is y= (3/2)x+ 2. If you don't like fractions, multiply by 2 to get 2y= 3x+ 2 which can be written in the same form as the line was originally given as 2y- 3x= 2.
 

What is the definition of perpendicular planes?

Perpendicular planes are two planes that intersect at a right angle, creating four right angles at the point of intersection.

How can I determine if two planes are perpendicular using their equations?

To determine if two planes are perpendicular, you can find the dot product of the normal vectors of the planes. If the dot product is equal to 0, then the planes are perpendicular.

What is the relationship between the slopes of perpendicular planes?

The slopes of perpendicular planes are negative reciprocals of each other. This means that if one plane has a slope of m, the other plane will have a slope of -1/m.

Can two planes be perpendicular if they are parallel to each other?

No, two planes cannot be perpendicular if they are parallel to each other. Perpendicular planes must intersect at a right angle, which is not possible if they are parallel.

How can I visualize if two planes are perpendicular?

You can visualize two planes as intersecting sheets of paper. If the two sheets are perpendicular, they will form a perfect "L" shape where they intersect.

Similar threads

MHB Math Cumulative Review: Find the equation of a Line
    • General Math
Replies
2
Views
2K
MHB Finding lines through given point perpendicular and parallel to given line
    • General Math
Replies
1
Views
924
B Existence of parallels in axiomatic plane geometries
    • General Math
2
Replies
36
Views
4K
B Vector perpendicular to a plane defined by two vectors
    • General Math
Replies
5
Views
2K
I How can you represent a point by "z = x + iy" as shown here?
    • General Math
Replies
10
Views
1K
I Finding vertex of a 3D Triangle on a Plane
    • General Math
Replies
3
Views
885
MHB Planes 1 & 2 Intersect: Find Scalar Equations
    • General Math
Replies
1
Views
1K
I What happens if I place light sources on the plane perpendicular to the principle axis of cylindrical lens?
    • Optics
Replies
1
Views
655
MHB Find the equation of the plane given a point and two planes
    • General Math
Replies
1
Views
2K
B How to calculate a perpendicular line?
    • General Math
Replies
11
Views
2K

Hot Threads

Recent Insights

Back
Top