Q.236. Reflection of a point in a plane

Question 236.
Find the image (reflection) of the point (2, 5, -1) in the plane x + y + x = 1.

Answer 236.
By reflection of the point in a plane is meant a point which is at the same distance from the plane but on the other side of the plane and the line joining the point and its reflection is perpendicular to the plane.

Normal to the plane x + y + z - 1 = 0 is
(1, 1, 1)
=> the relection of the point (2, 5, -1) in the plane x + y + z - 1 = 0 lies on the line
x - 2 = y - 5 = z + 1 = k, k ∈ R.
=> x = 2 + k,
y = 5 + k and
z = - 1 + k

Let for some k, it represent the foot of perpendicular from the point (2, 5, -1) to the plane
=> (2 + k) + (5 + k) + (-1 + k) - 1 = 0
=> k = - 5/3
=> foot of perpendicular is
[2-5/3, 5-5/3, -1-5/3] = (1/3, 10/3, - 8/3)

If (a, b, c) is the reflection of the point (2, 5, -1), the above foot of perpendicular is its midpoint.
=> (a+2)/2 = 1/3, (b+5)/2 = 10/3 and (c-1)/2 = - 8/3
=> a = - 4/3, b = 5/3 and c = - 13/3
=> the image of the point is
(-4/3, 5/3, -13/3).

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