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Length of the projection of a segment on a plane (distance between feet of perpendiculars)

This type of question, asked 3 times: 2021 Paper 4, 2021 Paper 5, 29 Jun 2022. All 3 questions below, oldest first.

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  1. Question 1

    2021 Paper 4

    61
    A plane passes through the points A(1,2,3),B(2,3,1)A(1,2,3),B(2,3,1) and C(2,4,2)C(2,4,2). If OO is the origin and PP is (2,−1,1)(2, - 1,1), then the projection of OP→\overrightarrow{OP} on this plane is of length:

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  2. Question 2

    2021 Paper 5

    63
    If for a>0a> 0, the feet of perpendiculars from the points A(a,−2a,3)A(a, - 2a,3) and B(0,4,5)B(0,4,5) on the plane lx+my+nz=0lx + my + nz = 0 are points C(0,−a,−1)C(0, - a, - 1) and DD respectively, then the length of line segment CDCD is equal to:

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  3. Question 3

    29 Jun 2022

    83
    Let dd be the distance between the foot of perpendiculars of the points P(1,2,−1)P(1,2,-1) and Q(2,−Q(2, - 1,3) on the plane −x+y+z=1-x+y+z= 1. Then d2d^{2} is equal to