How do you find parametric equations for the line of intersection of two planes 2x - 2y + z = 1, and 2x + y - 3z = 3?

Answer 1

#vec r = ((7/6), (2/3), (0)) + lambda ((5), (8), (6))#

for the line, we will need

A) a point that it actually passes through, say #vec a#, AND

B) a vector describing the direction in which it travels, say #vec b#

....such that the line itself is

#vec r = vec a + lambda vec b qquad square#

For #vec a#, we can simply choose the completely arbitrary point, so here we choose the point at which z = 0

This means that the plane equations, namely

#pi_1: 2x - 2y + z = 1#

#pi_2: 2x + y - 3z = 3#

become
#2x - 2y = 1#
#2x + y = 3#

and these we solve as simultaneous equations to get

#x = 7/6, y = 2/3# and of course #z = 0#

so #vec a = ((7/6), (2/3), (0))#

for #vec b# we need to calculate the vector cross product of the normal vectors for #pi_1# and #pi_2#.

In the drawing below, we are looking right down the line of intersection, and we get an idea as to why the cross product of the normals of the red and blue planes generates a third vector, perpendicular to the normal vectors, that defines the direction of the line of intersection.

for a generalised plane #pi: ax + by + cz = d#, the normal vector is: #vec n = ((a), (b), (c))#

so for #pi_1: 2x - 2y + z = 1#, the normal vector is #vec n_1 =((2), (-2), (1))#

and for #pi_2: 2x + y - 3z = 3#, the normal vector is #vec n_2 = ((2), (1), (-3))#

the cross product #vec b = vec n_1 times vec n_2# is the following determinant:

#vec b = det [(hat i, hat j, hat k),(2, -2, 1), (2, 1, -3) ]#

# = ((5), (8), (6))#

so we combine this all as indicated in #square# as follows

#vec r = ((7/6), (2/3), (0)) + lambda ((5), (8), (6))#

There are an infinite number of ways of expressing this line. #vec a# can be any point on the line. And the direction vector # vec b # can be any multiple of the one given herein.

The important bit, I guess is the method.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find parametric equations for the line of intersection of two planes, you need to solve the system of equations formed by the two planes. Here are the steps:

  1. Write down the equations of the planes: Plane 1: 2x - 2y + z = 1 Plane 2: 2x + y - 3z = 3

  2. Choose two variables to eliminate. In this case, let's choose to eliminate x.

  3. Solve one of the equations for x in terms of the other variables. From Plane 1: 2x = 2y - z + 1 x = y - 0.5z + 0.5

  4. Substitute the expression for x into the other equation. From Plane 2: 2(y - 0.5z + 0.5) + y - 3z = 3

  5. Simplify and solve for one variable. In this case, solve for y: 2y - z + 1 + y - 3z = 3 3y - 4z = 2 y = (2 + 4z) / 3

  6. Now, use the equation obtained for y and the expression for x to form parametric equations. Let z be the parameter: x = y - 0.5z + 0.5 x = ((2 + 4z) / 3) - 0.5z + 0.5

  7. Finally, express z as the parameter and write down the parametric equations: z = t (where t is the parameter) x = ((2 + 4t) / 3) - 0.5t + 0.5 y = ((2 + 4t) / 3)

These parametric equations represent the line of intersection of the two planes.

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7