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Let the point A'(3, 1, 4) be the mirror image of the point A(1, -1, 2) in a line L . If the line L passes through the point P(5, -2, ) , then the value of is equal to _______

Correct answer

2

Step-by-step solution

The line L acts as the perpendicular bisector of the line segment joining a point and its mirror image. First, we find the midpoint M of the segment AA' : M = ( 1+3 2 , -1+1 2 , 2+4 2 ) = (2, 0, 3) Since M lies on the line L and P(5, -2, ) also lies on L , the direction ratios of the line L are proportional to the vector MP : MP = (5 - 2) i + (-2 - 0) j + ( - 3) k = 3 i - 2 j + ( - 3) k The direction ratios of the line segment AA' are: AA' = (3 - 1) i + (1 - (-1)) j + (4 - 2) k = 2 i + 2 j + 2 k Since the line L is

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