The plane P1 has equation r = (2,1,1)+lambda(-2,1,8)+mu(1,-3,-9)
The plane P2 has equation r = (2,0,1)+s(1,2,1)+t(1,1,1)
Show that for points that lie on P1 and P2, lambda=mu
I have no idea how to start... help! Thanks!
I will use l for lambda and m for mu
Plane1: (x,y,z) = (2,1,1) + l(-2,1,8) + m(1,-3,-9)
Plane2: (x,y,z) = (2,0,1) + s(1,2,1) + t(1,1,1)
You want the x's, the y's and the z's to be the same
2 - 2l + m = 2 + s + t
s + t + 2l - m = 0 , #1
1 + l - 3m = 0 + 2s + t
2s + t - l + 3m = 1 , #2
1 + 8l - 9m = 1 + s + t
s+ t - 8l + 9m = 0 , #3
#1 - #3:
10l - 10m = 0
l = m
lamda = mu
Now that wasn't bad, perhaps a bit lucky the way it worked out
To show that for points that lie on P1 and P2, lambda equals mu, we need to find the values of lambda and mu that satisfy both plane equations simultaneously.
Let's first write the equations of both planes in vector form:
Plane P1: r = (2,1,1) + lambda(-2,1,8) + mu(1,-3,-9)
Plane P2: r = (2,0,1) + s(1,2,1) + t(1,1,1)
Now, let's equate the two plane equations:
(2,1,1) - lambda(2,-1,-8) - mu(1,3,9) = (2,0,1) + s(1,2,1) + t(1,1,1)
To compare the coefficients of lambda on both sides, we can set up the following equation for the x-coordinate:
2 - lambda(2) + mu(1) = 2 + s(1) + t(1)
Simplifying this equation, we get:
2 - 2lambda + mu = 2 + s + t
Rearranging terms, we have:
-2lambda + mu = s + t
Similarly, we can set up equations for the y-coordinate and z-coordinate:
-1 + lambda(1) - 3mu = 0 + 2s + t
8 - lambda(8) - 9mu = 1 + s + t
Simplifying these equations, we get:
lambda - 3mu = 2s + t (Equation 1)
-8lambda - 9mu = s + t (Equation 2)
To get lambda = mu, we need to eliminate lambda from these two equations. Multiplying Equation 1 by -8 and Equation 2 by 2, we get:
-8(lambda - 3mu) = -16s - 8t
-16lambda - 18mu = 2s + 2t
Adding this new equation to Equation 2, we eliminate lambda:
(-8lambda - 9mu) + (-16lambda - 18mu) = (s + t) + (2s + 2t)
-24lambda - 27mu = 3s + 3t
Dividing both sides by -3, we get:
8lambda + 9mu = -s - t (Equation 3)
Now, let's compare Equation 3 with the original Equation 2:
-8lambda - 9mu = s + t
8lambda + 9mu = -s - t
Comparing the two equations, we can see that the coefficients of lambda and mu are the negatives of each other. That means, if lambda = mu, then -lambda = -mu, and the two equations become:
-lambda = -mu
-lambda = -mu
Therefore, for points that lie on P1 and P2, lambda will indeed equal mu.
To summarize, we showed that for points that satisfy both plane equations P1 and P2, the values of lambda and mu are such that lambda = mu. We derived this result by equating the coefficients of the vector equations of the planes and eliminating lambda.