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Where L intercepts P

let P be the plane in R3 given by x + 2y + z = 0. Let L be the line through two points (1,0,1) and (0,1,1). Give the coordinates of the point where L intercepts P. 
 
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1 Answer

Hello Harold,
 
I'd solve this one by finding the parametric representation of the line passing through points (1, 0, 1) and (0, 1, 1). Starting at the point (1, 0, 1) and moving toward point (0, 1, 1), we get:
 
     x = 1 - t
     y = 0 + tt
     z = 1
 
Then, I'd plug these values in to the equation of the plane and solve for t:
 
     (1 - t) + 2(t) + 1 = 0
     t + 2 = 0
     t = -2
 
Then go back to the parametric representation of the line and plug in t = -2 to find the interception coordinates:
 
     x = 1 - (-2) = 3
     y = -2
     z = 1
 
And there we have it! The plane and the line intersect at the point (3, -2, 1).