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Three Dimensional Geometry

Short 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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Long 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.

Let l, m, n be the direction cosines of the given line PQ. Then

straight m equals space cos space 120 degree space equals space cos space left parenthesis 180 degree space minus space 160 degree right parenthesis space equals space minus cos space 60 degree space equals space minus 1 half. straight n space equals space cos space 135 space degree space equals space cos space left parenthesis 180 degree minus 45 degree right parenthesis space equals space minus cos space 45 degree space equals negative fraction numerator 1 over denominator square root of 2 end fraction

Now,

straight l squared plus straight m squared plus straight n squared space equals space 1 space space space space space space space space space rightwards double arrow space space space space straight l squared plus 1 fourth plus 1 half space equals 1 space space space space rightwards double arrow space space space straight l squared space equals space 1 fourth

therefore space space space straight l space equals space 1 half

[∵ line PQ makes acute angle with x-axis]

therefore

direction cosines of PQ are

1 half comma space minus 1 half space minus fraction numerator 1 over denominator square root of 2 end fraction

Now A and B are (–1, 2, 1), (4, 3, 5) respectively

therefore

projection of AB on PQ =

left parenthesis 4 plus 1 right parenthesis. space open parentheses 1 half close parentheses plus space left parenthesis 3 minus 2 right parenthesis space open parentheses negative 1 half close parentheses plus left parenthesis 5 minus 1 right parenthesis. space open parentheses negative fraction numerator 1 over denominator square root of 2 end fraction close parentheses

open square brackets because space space of space left parenthesis straight x subscript 2 minus straight x subscript 1 right parenthesis space straight l space plus 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 subscript 1 right parenthesis space straight n close square brackets

equals negative 5 over 2 minus 1 half minus fraction numerator 4 over denominator square root of 2 end fraction space equals space 2 minus square root of 2

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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.

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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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Short 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.

135 Views