$Let A (- 1, 3, 3), B (3, 2, 1) then the projection of which point of the following on the$ $straight line through A and B falls on the line segment \bar{A B} (end points included)$

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$(1, - 1, 2)$

$(- 1, 1, 2)$

$(2, 1, - 1)$

$(2, - 1, 1)$

answer is A, B, D.

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Detailed Solution

$\begin{array}{l} For the option A \\ Equation of the plane passing through the point (1, - 1, 2) \\ and having AB as normal is 4 x - y - 2 z - 1 = 0 \\ π_{A} \cdot π_{B} < 0 \\ For the option B \\ Equation of the plane passing through the point (- 1, 1, 2) \\ and having AB as normal is 4 x - y - 2 z + 10 = 0 \\ π_{A} \cdot π_{B} < 0 \\ For the option C \\ Equation of the plane passing through the point (2, 1, - 1) \\ and having AB as normal is 4 x - y - 2 z - 9 = 0 \\ π_{A} \cdot π_{B} > 0 \\ For the option D \\ Equation of the plane passing through the point (2, - 1, 1) \\ and having AB as normal is 4 x - y - 2 z - 7 = 0 \\ π_{A} \cdot π_{B} < 0 \end{array}$