A and B are the points (2, – 1, 3) and (4, 2, 5). Find the pro
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    Three Dimensional Geometry

    Multiple Choice QuestionsShort Answer Type

    21.

    Find the projection of the line segment joining the points (1, 2, 3), (4, 3, 1) on the line with direction ratios 3, –6, –2.

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    22. Find the projection of the line segment joining the points (2, -3, 0), (0, 4, 5) on the line with direction cosines 2 over 3 comma space 3 over 7 comma space minus 6 over 7
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    23.

    A directed line segment makes angles 45° and 60° with x-axis and y-axis and an acute angle with z-axis. If P (– 1, 2, – 3) and Q (4, 3, 1) are two points in space, find the projection of PQ on the given line.

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    Multiple Choice QuestionsLong Answer Type

    24. If A and B are the points (– 1, 2, 1) and (4, 3, 5). Find the projection of AB on a line which makes angles of 120° and 135° with y-axis and z-axis respectively and acute angle with x-axis.
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    25. A and B are the points (2, – 1, 3) and (4, 2, 5). Find the projection of AB on a line which is inclined at equal acute angles with the co-ordinate axes.

    Let α be the angle which the line PQ makes with all the axes.
    therefore  its direction cosines are cos space straight alpha comma space space cosα comma space cosα.
    therefore space space cos squared straight alpha plus cos squared straight alpha plus cos squared straight alpha space equals space 1 space space space space space space space space space left square bracket straight l squared plus straight m squared plus straight n squared space equals space 1 right square bracket
    therefore space space 3 space cos squared space straight alpha space equals space 1 space space space space rightwards double arrow space space space space cos squared straight alpha space equals space 1 third space space space rightwards double arrow space space space cos space straight alpha space equals space plus-or-minus fraction numerator 1 over denominator square root of 3 end fraction
    therefore    cos space straight alpha space equals space fraction numerator 1 over denominator square root of 3 end fraction                        open square brackets because space line space makes space acute space angle space straight alpha space with space each space axis close square brackets
    therefore                  straight l space equals straight m space equals space straight n space space equals fraction numerator 1 over denominator square root of 3 end fraction
    where l, m, n are direction cosines of line PQ.
    A, B are points (2, – 1, 3), (4, 2, 5).
    therefore    projection of AB on PQ = (4 - 2). fraction numerator 1 over denominator square root of 3 end fraction + (2+1).  fraction numerator 1 over denominator square root of 3 end fraction plus left parenthesis 5 minus 3 right parenthesis. space fraction numerator 1 over denominator square root of 3 end fraction
                                                       open square brackets because space space of space space left parenthesis straight x subscript 2 minus straight x subscript 1 right parenthesis straight l space plus space left parenthesis straight y subscript 2 minus straight y subscript 1 right parenthesis space straight m space plus space left parenthesis straight z subscript 2 minus straight z right parenthesis space straight n close square brackets
                                 equals space fraction numerator 2 over denominator square root of 3 end fraction plus fraction numerator 3 over denominator square root of 3 end fraction plus fraction numerator 2 over denominator square root of 3 end fraction space equals space fraction numerator 2 plus 3 plus 2 over denominator square root of 3 end fraction space equals fraction numerator 7 over denominator square root of 3 end fraction space units. space
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    26.

    If P, Q, R, S are the points (– 2, 3, 4), (– 4, 4, 6), (4, 3, 5), (0, 1, 2), prove by projection that PQ is perpendicular to RS.

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    27. Show that the line through the points (1, –1, 2), (3, 4, –2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
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    Multiple Choice QuestionsShort Answer Type

    28. A, B, C, D are the points (1, – 1, 0), (2, 1, – 1), (– 3, 2, 2) and (0, – 2, – 1) respectively. Find the projection of AB on CD.
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    29. The projection of a line segment on x, y and z-axes are respectively 12. 4 and 3. Find the length and the direction-cosines of the line segment.
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    30.

    The projections of a directed line segment on the co-ordinate axes are 6, -3, 2. Find its length and direction cosines. 

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