- Determine if the point (4,-2,3) lies in the plane with vector equation
*(x, y, z) = (2, 0, -1) + s(4, -2, 1) + t(-3, -1, 2).*

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Let (x,y,z) = (4,-2,3). Rearrange the set of three equations to give

1) 4s - 3t + 2 = 4 implies 4s - 3t = 2

2) -2s - t + 0 = -2 implies -2s - t = -2

3) s + 2t -1 = 3 implies s + 2t = 4

Add 1) and 2) together giving

2s - 4t = 0 implies s = 2t

Putting this in to 3) and solving gives t = 1 (and hence s = 2)

Check this against equation 1). 4(2) - 3(1) = 5

This does not give the correct result, **therefore the point (4,-2,3) does not lie on the plane defined by the vector equation.**

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