How do you find parametric equations for the line of intersection of two planes 2x  2y + z = 1, and 2x + y  3z = 3?
for the line, we will need
A) a point that it actually passes through, say
B) a vector describing the direction in which it travels, say
....such that the line itself is
For
This means that the plane equations, namely
become
and these we solve as simultaneous equations to get
so
for
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
so for
and for
the cross product
so we combine this all as indicated in
There are an infinite number of ways of expressing this line.
The important bit, I guess is the method.
By signing up, you agree to our Terms of Service and Privacy Policy
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:

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

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

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

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

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

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

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.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a onesided 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 onesided 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 onesided 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 onesided 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.
 How do you differentiate the following parametric equation: # x(t)=1lnt, y(t)= costt^2 #?
 What is the arc length of #f(t)=(3te^t,te^t) # over #t in [2,4]#?
 For #f(t)= (sin^2t,t/pi2)# what is the distance between #f(pi/4)# and #f(pi)#?
 What is the arclength of #(t^2t,t^21)# on #t in [1,1]#?
 What is the arclength of #(t^2lnt,(lnt)^2)# on #t in [1,2]#?
 98% accuracy study help
 Covers math, physics, chemistry, biology, and more
 Stepbystep, indepth guides
 Readily available 24/7