Problem 35 - Calculus Early Transcendentals

Question

Find an equation of the line of intersection of the planes \(Q\) and \(R\). $$Q:-x+2 y+z=1 ; R: x+y+z=0$$

Step-by-Step Solution

Answer

Question: Find the equation of the line of intersection of the two planes Q: -x + 2y + z = 1 and R: x + y + z = 0. Answer: The line of intersection of the two planes can be represented parametrically as L: (x, y, z) = (1/3 - t, -1/3, t).

Step 1: Express the intersection line parametrically

Let the intersection of the two planes be the line L. Set z = t, thus the line L can be expressed as (x, y, t).

Step 2: Substitute the parameter into the plane equations

Substitute z=t into plane Q and plane R, we then get the following two equations: 1. -x+2y+t=1 2. x+y+t=0

Step 3: Solve for x and y in terms of t

Now, add these two equations together to eliminate x, we get: 3y = -1 Then, solve for y, we get y = -1/3. Now, use equation 2 to solve for x: x = -y - t = -(-1/3) - t = 1/3 - t.

Step 4: Write the parametric form of the line of intersection

We now have x, y and z all in terms of t. So the intersection line L can be expressed as follows: L: (x, y, z) = (1/3 - t, -1/3, t) This is the equation of the line of intersection of the two planes in parametric form. The intersection line is a one dimensional line in a three dimensional space, which can be expressed as a function of one variable t.