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Homework Statement
Find the parametric and canonical equation of the line L passing through the points A = [1, 0, 2] and B = [3, 1, −2]; check whether the point M = [7, 3, 1] lies on L.
Homework Equations
Canonical equation of a line in space
x-x0 / l = y-y0 / m = z-z0 / n
Parametric equation of a line
x=lt+x0
y=mt+y0
z=nt+z0
where x0 y0 z0 are coordinates of a point sitting on the line
l,m,n are coordinates of the direction of the line
The Attempt at a Solution
The vector AB = B - A = [3, 1, −2] - [1, 0, 2] = [2,1,-4]
To see if M sits on the line we sub in it's coordinates into both equations:
Canonical =
7-1 / 2 = 3-0/1 = 1-2/4
6/2=3/1≠-1/-4
Parametric
7 = 2t+1
3 = 1t+0
1 = -4t+2
6=2t
3=1t
-1=-4t
Because we have an inequaity / inconsistency in the z component we can conclude that the point M = [7,3,1] does not lie on the line L. For M to lie on the line L all three equations would have to equal each other in canonical form, and in parametric form all values of t would have to be the same.
Just wondering if my conclusion is correct? Thanks :)