Perpendicular planes, solve for parameter a

In summary, the problem requires determining for what values of the parameter a the given planes in R3 are perpendicular. This can be done by finding the dot product of the two normal vectors of the planes and setting it equal to zero. Using the equations given, the normal vectors for each plane are found to be (a,0,-5) and (a,a,5). Setting their dot product equal to zero gives a^2 - 25 = 0, leading to a = -5 or 5. Therefore, the planes are perpendicular for values of a = {-5,5}.
  • #1
concon
65
0

Homework Statement


Determine for what value/s of the parameter a the following planes in R3 are perpendicular:
ax + 0y - 5z = 3
ax + ay + 5z = -26
Write answer in form {a,b} or {a}



Homework Equations


I know that in R2 two vectors are perpendicular if
u*v = 0
What what do I use for R3 with planes?



The Attempt at a Solution


To solve this problem I need to know what it means for two planes to be perpendicular. I know in R2 that means there dot product is zero, what does that mean in R3? Also dot product?
 
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  • #2
concon said:

Homework Statement


Determine for what value/s of the parameter a the following planes in R3 are perpendicular:
ax + 0y - 5z = 3
ax + ay + 5z = -26
Write answer in form {a,b} or {a}



Homework Equations


I know that in R2 two vectors are perpendicular if
u*v = 0
What what do I use for R3 with planes?



The Attempt at a Solution


To solve this problem I need to know what it means for two planes to be perpendicular. I know in R2 that means there dot product is zero, what does that mean in R3? Also dot product?
Two planes are perpendicular if their normals are perpendicular. That can be determined by getting the dot product of the two normals.
 
  • #3
You need to know:

- what a vector perpendicular to a plane is,
- how to find a vector perpendicular to a plane (and why)
- that two planes are perpendicular when vectors perpendicular (normal) to them are perpendicular (and why)

then you can proceed easily (knowing what you do).

Try to think more precisely!
To solve this problem I need to know what it means for two planes to be perpendicular. I know in R2 that means there dot product is zero
In R2 there is no plane, there are lines.

Do you know how to check if two line in R2 are perpendicular?
For example: when are these lines perpendicular?

ax - 5 y= 1458966.1973
ax + 5 y= -125896.5586

what does that mean in R3? Also dot product?

dot product of what? That's the point!
 
  • #4
maajdl said:
You need to know:

- what a vector perpendicular to a plane is,
- how to find a vector perpendicular to a plane (and why)
- that two planes are perpendicular when vectors perpendicular (normal) to them are perpendicular (and why)

then you can proceed easily (knowing what you do).

Try to think more precisely!

In R2 there is no plane, there are lines.

Do you know how to check if two line in R2 are perpendicular?
For example: when are these lines perpendicular?

ax - 5 y= 1458966.1973
ax + 5 y= -125896.5586



dot product of what? That's the point!

Okay I got the solution: {-5,5}

Normal of the first equation n = (a,0,-5)
Normal of the second equation m = (a,a,5)

thus n*m = a^2 -25 = 0
thus a = -5 or 5

Thanks to both of y'all for the replies. It helped a lot and I appreciate it!
 

Related to Perpendicular planes, solve for parameter a

1. What are perpendicular planes?

Perpendicular planes are two planes that intersect at a 90 degree angle, forming a right angle.

2. How do you determine if two planes are perpendicular?

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

3. What is the equation for finding the parameter a in perpendicular planes?

The equation for finding the parameter a in perpendicular planes is a = -[n1 * (p2 - p1)] / n1 * n2, where n1 and n2 are the normal vectors of the planes, and p1 and p2 are any points on the planes.

4. 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 90 degree angle.

5. How many solutions can there be for a when solving for perpendicular planes?

There can be infinitely many solutions for a when solving for perpendicular planes. This is because there are infinitely many points that can be chosen on the planes to use in the equation for finding a.

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