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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.

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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.

Let l, m, n be the direction-cosines of the line and let the length of the line segment be r.
∵ projections of the line segment on the axis are 12, 4, 3,
∴ l r = 12, m r = 4, n r = 3
Squaring and adding, we get,
(l² + m² + n²) r² = 144 + 16 + 9
∴ (1) (r)² = 169, ∴ r = 13
$therefore$ $straight l space equals 12 over 13 comma space space straight m space equals 4 over 13 comma space straight n space equals 3 over 13$

∴ length of the line = 13 units
and direction-cosines of line are $12 over 13 comma space 4 over 13 comma space 3 over 13. comma space$

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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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