SOLUTION: Where does the line through (1, 2, 1) and (2, 1, 4) intersect the plane 2x + 3y + z = 10?

Question 1203546: Where does the line through (1, 2, 1) and (2, 1, 4) intersect the plane 2x + 3y + z = 10?
Answer by math_tutor2020(3816) (Show Source): You can put this solution on YOUR website!

A = (1, 2, 1)
B = (2, 1, 4)
v = direction vector from A to B
v = B - A
v = (2,1,4) - (1,2,1)
v = (2-1,1-2,4-1)
v = (1,-1,3)

To go from A to B we must:

Go 1 unit along the positive x axis.
Go 1 unit along the negative y axis.
Go 3 units along the positive z axis.

The direction vector in 3D settings is similar to the concept of slope in 2D situations. Both tell us how to get from one point to another on the same line.

One possible equation for the line AB is (x,y,z) = (1+t,2-t,1+3t)

The scratch work to figure out the line is shown below.
(x,y,z) = startPoint + directionVector*t
(x,y,z) = (1,2,1) + (1,-1,3)*t
(x,y,z) = (1,2,1) + (t,-t,3t)
(x,y,z) = (1+t,2-t,1+3t)

The variable t represents the moment in time. Example: t = 5 means 5 seconds
t = 0 leads to (x,y,z) = (1,2,1) while t = 1 leads to (2,1,4)
This will help verify that we did things correctly.
I leave the scratch work calculations for the student to do.

Other equations for line AB are possible.
For instance, we could have made the start point (2,1,4).
The direction vector could also be scaled up or down.

Because (1+t,2-t,1+3t) represents the location along line AB, at some time t, we can plug those x,y,z coordinates into the equation of the plane.
This will ensure that we find where the line and plane intersect (if such a thing happens).

2x + 3y + z = 10
2(1+t) + 3(2-t) + (1+3t) = 10
2+2t + 6-3t + 1+3t
2t+9 = 10
2t = 10-9
2t = 1
t = 1/2
t = 0.5

Use this time value to figure out where we are on the line.
(x,y,z) = (1+t,2-t,1+3t)
(x,y,z) = (1+0.5,2-0.5,1+3*0.5)
(x,y,z) = (1.5,1.5,2.5)

At time t = 0.5, we're at the location (1.5,1.5,2.5) which is interestingly the midpoint of segment AB.

Let's check to see if this point is on the plane.
2x + 3y + z = 10
2*1.5 + 3*1.5 + 2.5 = 10
3 + 4.5 + 2.5 = 10
3 + 7 = 10
10 = 10
The answer has been confirmed.

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Answer:
(1.5, 1.5, 2.5)
or its fraction equivalent (3/2, 3/2, 5/2)

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