Consider the following planes.
5x − 4y + z = 1, 4x + y − 5z = 5
a) Find parametric equations for the line of intersection of the planes.
b) Find the angle between the planes
mosey on over to
http://math.stackexchange.com/questions/475953/how-to-calculate-the-intersection-of-two-planes
and have a gander there.
a) To find the parametric equations for the line of intersection of the planes, we can solve the system of equations formed by setting the two plane equations equal to each other:
5x - 4y + z = 1 ---- (1)
4x + y - 5z = 5 ---- (2)
To eliminate one variable, we can multiply equation (1) by 4 and equation (2) by 5:
20x - 16y + 4z = 4 ---- (3)
20x + 5y - 25z = 25 ---- (4)
Now, subtract equation (3) from equation (4) to eliminate x:
20x - 16y + 4z -(20x + 5y - 25z) = 4 - 25
-21y + 29z = -21 ---- (5)
To express y in terms of z, we can isolate y in equation (5):
-21y = -29z - 21
y = (29z + 21) / 21
Similarly, to express x in terms of z, we can substitute y back into equation (1):
5x - 4((29z + 21) / 21) + z = 1
5x - (116z + 84) / 21 + z = 1
105x - 100z - 116z - 84 + 21z = 21
105x - 195z = 105
x = (195z + 105) / 105
Therefore, the parametric equations for the line of intersection of the planes are:
x = (195z + 105) / 105
y = (29z + 21) / 21
z is a parameter.
b) To find the angle between the planes, we need to find the angle between their normal vectors. The normal vector of a plane can be found by taking the coefficients of x, y, and z in the plane equation.
For plane 1:
5x - 4y + z = 1
The normal vector of plane 1 is <5, -4, 1>.
For plane 2:
4x + y - 5z = 5
The normal vector of plane 2 is <4, 1, -5>.
The angle between two vectors can be found using the dot product:
dot(N1, N2) = ||N1|| * ||N2|| * cos(theta),
where ||N1|| and ||N2|| are the magnitudes of the normal vectors N1 and N2, and theta is the angle between the planes.
The magnitudes of the normal vectors are:
||N1|| = sqrt(5^2 + (-4)^2 + 1^2) = sqrt(42)
||N2|| = sqrt(4^2 + 1^2 + (-5)^2) = sqrt(42)
The dot product of the two normal vectors is:
dot(N1, N2) = 5*4 + (-4)*1 + 1*(-5) = 20 - 4 - 5 = 11
Substituting these values into the formula, we have:
11 = sqrt(42) * sqrt(42) * cos(theta)
Simplifying:
11 = 42 * cos(theta)
Solving for theta:
cos(theta) = 11 / (42)
theta = arccos(11 / 42)
So, the angle between the planes is arccos(11 / 42).
To find the parametric equations for the line of intersection of the planes, we need to find two points on the line.
Step 1: Solve the system of equations formed by the planes.
We'll solve the system of equations:
5x - 4y + z = 1
4x + y - 5z = 5
Multiplying the first equation by 4 and the second equation by 5 to eliminate the y term, we get:
20x - 16y + 4z = 4
20x + 5y - 25z = 25
Subtracting the first equation from the second equation, we eliminate the x term:
21y - 29z = 21
Step 2: Set a parameter.
Let's set y = t, where t is a parameter.
Step 3: Express the remaining variables in terms of the parameter.
From the equation 21y - 29z = 21, substitute y = t to get:
21t - 29z = 21
We can solve for z in terms of t:
-29z = 21 - 21t
z = (21 - 21t) / -29
Step 4: Write the parametric equations.
We have:
x = (5 - 4y + z) / 5
x = (5 - 4t + (21 - 21t) / -29) / 5
x = (105 - 20t + 21 - 21t) / -145
x = (-40t + 126) / -145
x = (40t - 126) / 145
y = t
z = (21 - 21t) / -29
Therefore, parametric equations for the line of intersection of the planes are:
x = (40t - 126) / 145
y = t
z = (21 - 21t) / -29
Now, let's move on to finding the angle between the planes.
To find the angle between the planes, we can use the formula:
cos(theta) = (n1 · n2) / (||n1|| ||n2||)
Where n1 and n2 are the normal vectors to the planes.
Step 1: Find the normal vectors to the planes.
For plane 1: 5x - 4y + z = 1
The coefficients of x, y, and z in the equation represent the normal vector.
So, the normal vector to plane 1 is n1 = (5, -4, 1).
For plane 2: 4x + y - 5z = 5
The coefficients of x, y, and z in the equation represent the normal vector.
So, the normal vector to plane 2 is n2 = (4, 1, -5).
Step 2: Compute the dot product and the magnitudes.
n1 · n2 = 5 * 4 + (-4) * 1 + 1 * (-5) = 20 - 4 - 5 = 11
||n1|| = sqrt(5^2 + (-4)^2 + 1^2) = sqrt(25 + 16 + 1) = sqrt(42)
||n2|| = sqrt(4^2 + 1^2 + (-5)^2) = sqrt(16 + 1 + 25) = sqrt(42)
Step 3: Compute the angle.
cos(theta) = (n1 · n2) / (||n1|| ||n2||)
cos(theta) = 11 / (sqrt(42) * sqrt(42))
cos(theta) = 11 / 42
theta = acos(11 / 42)
So, the angle between the planes is equal to acos(11 / 42).