Linear Algebra. Intersection Between plane and line containing certain points.

Question

Find the point of intersection of the plane 2x + 3y − 5z = 10 with the line
containing the points (1, 1, 1) and (5, 0, 4).

Answer 1

Claire, I hope I can still help.

 

We need to find the parametric representation of our line. We use...

 

F(t) = <x₀, y₀, z₀> + t<x₁ - x₀, y₁ - y₀, z₁ - z₀>

       = <1, 1, 1> + t<5 - 1, 0 - 1, 4 - 1>

       = <1, 1, 1> + t<4, -1, 3>

       =< 1 + 4t, 1 - t, 1 + 3t>

 

Now, we take our "x, y, and z" and plug them into our equation of the plane.

 

2(1 + 4t) + 3(1 - t) - 5(1 + 3t) = 10

            2 + 8t + 3 - 3t - 5 - 15t = 10

                                           -10t = 10

                                                t = -1

 

Now we just need to plug in t to our parametric representation.

 

<1 + 4(-1), 1 - (-1), 1 + 3(-1)>

(-3, 2, -2) is our point.

 

Hope this helps!