Calculating Acute Angle Between Planes: Ax+By+Cz=D & Ex+Fy+Gz=H

Thread starter nk735
Start date Apr 8, 2008
Tags

Angle Planes

In summary, to calculate the acute angle between two planes, you can use the formula cosθ = |a1a2 + b1b2 + c1c2| / √(a1^2 + b1^2 + c1^2) * √(a2^2 + b2^2 + c2^2), where the coefficients of the two planes are plugged in and the inverse cosine is taken to find the angle. Alternatively, you can use the dot product of the normal vectors of the two planes to find the angle. The significance of this angle is that it determines the tilt or intersection of the two planes, and it is always a positive value.

Apr 8, 2008

nk735

Homework Statement

Find the acute angle between the planes Ax+By+Cz=D and Ex+Fy+Gz=H

Homework Equations

The Attempt at a Solution

I have no idea

Physics news on Phys.org

Apr 8, 2008

Dick

Science Advisor

Homework Helper

The angle between the planes is the same as the angle between the normals to the planes.

FAQ: Calculating Acute Angle Between Planes: Ax+By+Cz=D & Ex+Fy+Gz=H

How do you calculate the acute angle between two planes?

To calculate the acute angle between two planes, you will need to use the formula: cosθ = |a1a2 + b1b2 + c1c2| / √(a1^2 + b1^2 + c1^2) * √(a2^2 + b2^2 + c2^2), where a1, b1, and c1 are the coefficients of the first plane (Ax+By+Cz=D) and a2, b2, and c2 are the coefficients of the second plane (Ex+Fy+Gz=H). Once you have the value of cosθ, you can use the inverse cosine function to find the acute angle between the two planes.

Can you provide an example of calculating the acute angle between two planes?

Sure, let's say we have two planes: 2x + 3y - z = 7 and 4x - 5y + 2z = 10. We will first need to find the coefficients for both planes, which are: a1 = 2, b1 = 3, c1 = -1 and a2 = 4, b2 = -5, c2 = 2. Plugging these values into the formula above, we get: cosθ = |2*4 + 3*(-5) + (-1)*2| / √(2^2 + 3^2 + (-1)^2) * √(4^2 + (-5)^2 + 2^2) = 0.16. Taking the inverse cosine of 0.16, we get the acute angle between the two planes to be approximately 82.2 degrees.

What is the significance of finding the acute angle between two planes?

The acute angle between two planes determines how much they tilt or intersect with each other. A smaller angle means the planes are closer to being parallel, while a larger angle indicates they are more perpendicular to each other. This information can be useful in various fields such as engineering, physics, and mathematics.

Can the acute angle between two planes be negative?

No, the acute angle between two planes is always a positive value. This is because the formula for calculating the angle takes the absolute value of the dot product of the two plane vectors, making the result always positive.

Is there a faster way to calculate the acute angle between two planes?

Yes, instead of using the formula, you can also use the dot product of the normal vectors of the two planes. The normal vectors are simply the coefficients of the planes (a, b, c) normalized to a unit vector. The dot product of two unit vectors is equal to the cosine of the angle between them, so you can simply take the inverse cosine of the dot product to find the acute angle between the planes.