Determine the interaction of the line of intersection of the planes x + y - z = 1 and 3x + y + z = 3 with the line of intersection of the planes 2x - y + 2z = 4 and 2x + 2y + z = 1.
Find the line of intersection of the first two planes, using the method I showed you yesterday.
Do the same for the second pair of planes.
Write each line in parametric form but use a different parameter, use t in the first one, k in the second
Now there are three possibilities.
1. the two lines are parallel
2. The two lines intersect
3. the two lines miss each other, and are not parallel
(A fourth trivial case would be if the two lines end up the same line)
for #1, look at the direction numbers. Are they the same?
If not set the x's and the y's of the two sets of parametric equations equal to each other, you should be able to solve for t and k
If the values of t and k also satisfy the parametric equations for z, then they actually intersect in a point, if not, they will miss each other.
(How about letting me know if you are getting this stuff. I have now helped you with about 6 of these vector problems but you have not replied if you are following this .)
Well, let's break this down step by step. First, we need to find the line of intersection of the planes x + y - z = 1 and 3x + y + z = 3. To do this, we can use the method of substitution.
By solving these two equations simultaneously, we find that the line of intersection is:
x = 1
y = 0
z = 0
Now, let's find the line of intersection of the planes 2x - y + 2z = 4 and 2x + 2y + z = 1. Again, using the method of substitution, we can solve these two equations and find:
x = 1
y = -1
z = 0
So, we have two lines:
First line: x = 1, y = 0, z = 0
Second line: x = 1, y = -1, z = 0
Now, let's determine the interaction of these two lines. Ahh, this reminds me of a classic comedy routine - two lines walk into a bar. The bartender looks at them and says, "Sorry, we don't serve lines here!" The first line shrugs and says, "That's okay, we're just here for a good time." Ba dum tss!
In all seriousness, since both lines have the same x and z values, but different y values, we can conclude that these two lines are skew lines. They never meet or intersect in 3D space. It's like two friends heading in different directions - they may be close, but they're not on the same path.
To determine the interaction of the line of intersection of two planes, we need to first find the equations of both planes and then find the line of intersection for each pair of planes. Finally, we will find the interaction of these two lines.
Let's start by finding the equation of the first pair of planes.
Equation of Plane 1: x + y - z = 1 (Equation A)
Equation of Plane 2: 3x + y + z = 3 (Equation B)
To find the line of intersection for these two planes, we need to solve the system of equations formed by these two planes. We can do that by applying the method of elimination.
First, multiply Equation A by 3:
3x + 3y - 3z = 3 (Equation C)
Now, let's subtract Equation B from Equation C to eliminate the y term:
(3x + 3y - 3z) - (3x + y + z) = 3 - 3
2y - 4z = 0
Divide through by 2:
y - 2z = 0 (Equation D)
We have eliminated the x term and obtained Equation D. Notice that Equation D does not have an x term, indicating that the line of intersection of these two planes lies in the xz plane.
Now let's find the equation of the second pair of planes.
Equation of Plane 3: 2x - y + 2z = 4 (Equation E)
Equation of Plane 4: 2x + 2y + z = 1 (Equation F)
To find the line of intersection for these two planes, we apply the method of elimination.
Let's multiply Equation E by 2:
4x - 2y + 4z = 8 (Equation G)
Now, subtract Equation F from Equation G to eliminate the x term:
(4x - 2y + 4z) - (2x + 2y + z) = 8 - 1
2x - 3y + 3z = 7 (Equation H)
We have eliminated the x term and obtained Equation H. Notice that Equation H does not have an x term, indicating that the line of intersection of these two planes lies in the yz plane.
Now, let's determine the interaction between these two lines.
From Equations D and H, we can write:
y - 2z = 0 (Equation I)
2x - 3y + 3z = 7 (Equation J)
Now, we need to solve these two equations simultaneously to find the values of x, y, and z.
From Equation I, we can isolate y:
y = 2z (Equation K)
Substituting Equation K into Equation J:
2x - 3(2z) + 3z = 7
2x - 6z + 3z = 7
2x - 3z = 7
Divide through by 2:
x - (3/2)z = 7/2
x = (3/2)z + 7/2 (Equation L)
We have expressed x in terms of z.
Finally, let's express x, y, and z in terms of a parameter t.
Using Equations K and L, we have:
x = (3/2)z + 7/2
y = 2z
Choosing z as the parameter t, we can write:
x = (3/2)t + 7/2
y = 2t
z = t
Therefore, the line of intersection of the planes x + y - z = 1 and 3x + y + z = 3 intersects the line of intersection of the planes 2x - y + 2z = 4 and 2x + 2y + z = 1 at the point (x, y, z) = ((3/2)t + 7/2, 2t, t).
To determine the interaction of the line of intersection of two planes with another line of intersection of two planes, we need to find the equations of both lines first. Let's start by finding the equations of the lines.
1. Finding the line of intersection of the planes x + y - z = 1 and 3x + y + z = 3:
To find the line of intersection, we need to solve the system of equations formed by the two planes. Let's do that:
Step 1: Write the system of equations:
x + y - z = 1 ----(1)
3x + y + z = 3 ----(2)
Step 2: Eliminate one variable:
To eliminate the variable 'y', we can multiply equation (1) by 3 and subtract it from equation (2):
(3x + y + z) - 3(x + y - z) = 3 - 3
3x + y + z - 3x - 3y + 3z = 0
-2y + 4z = 3
Step 3: Solve for remaining variables:
We have one equation with two variables, which means we can express one of the variables in terms of the other. Let's express 'y' in terms of 'z':
-2y = -4z + 3
y = (4z - 3) / 2
Substituting this value of 'y' back into equation (1):
x + (4z - 3) / 2 - z = 1
2x + 4z - 3 - 2z = 2
2x + 2z - 3 = 2
2x + 2z = 5
x + z = 5/2
x = (5/2) - z
So, the line of intersection of the planes x + y - z = 1 and 3x + y + z = 3 is given by the parametric equations:
x = (5/2) - t ----(3)
y = (4t - 3) / 2 ----(4)
z = t ----(5)
where 't' is a parameter.
2. Finding the line of intersection of the planes 2x - y + 2z = 4 and 2x + 2y + z = 1:
Again, we need to solve the system of equations formed by the two planes. Let's do that:
Step 1: Write the system of equations:
2x - y + 2z = 4 ----(6)
2x + 2y + z = 1 ----(7)
Step 2: Eliminate one variable:
To eliminate the variable 'y', we can multiply equation (6) by 2 and subtract it from equation (7):
(2x + 2y + z) - 2(2x - y + 2z) = 1 - (2 * 4)
2x + 2y + z - 4x + 2y - 4z = 1 - 8
-2x + 4z = -7
Step 3: Solve for remaining variables:
We have one equation with two variables, which means we can express one of the variables in terms of the other. Let's express 'x' in terms of 'z':
-2x = -4z - 7
x = (4z + 7) / 2
x = 2z + 7/2
Substituting this value of 'x' back into equation (6):
2(2z + 7/2) - y + 2z = 4
4z + 7 - y + 2z = 4
6z - y = 4 - 7
6z - y = -3
y = 6z + 3
So, the line of intersection of the planes 2x - y + 2z = 4 and 2x + 2y + z = 1 is given by the parametric equations:
x = 2z + 7/2 ----(8)
y = 6z + 3 ----(9)
z = t ----(10)
where 't' is a parameter.
Now that we have obtained the parametric equations for both lines of intersection, we can determine their interaction by solving the system formed by equations (3)-(5) and (8)-(10) simultaneously.