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Three Dimensional Geometry

Short Answer Type

11. Find the direction ratios and the direction cosines of the lines joining the pairs of points:
(– 1, –1, –1), (2, 3, 4)

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12. Find the direction cosines of the line passing through the two points (–2, 4, –5) and (1, 2, 3).

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Long Answer Type

13. Find the direction cosines of the sides of the triangle whose vertices are (3, 5, –4), (–1, 1, 2) and (–5, –5, –2).

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Short Answer Type

14.

Find the direction cosines of the line joining the points (2, 1, 2) and (4, 2, 0).

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15.

A line makes angles of 45° and 60° with the positive axes of x and y respectively. What angle does it make with the positive axis of z?

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16. If α, β, γ are the angles which a line makes with the axes, prove that sin²α + sin² β + sin²γ = 2.

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17.

Show that the points (2, 3, 4), (–1, –2, 1), (5, 8, 7) are collinear.

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18. Show that the points (–1, 2, –3), (4, 5, 1) and (9, 8, 5) are collinear.

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19. Show that the points A(2, 3, – 4), B(1, –2, 3) and C (3, 8, –11) are collinear.

The given points are A(2, 3, – 4), B(1, –2, 3), C(3, 8, –11)
The direction ratios of the AB are 1 – 2, – 2 – 3, 3 + 4 i.e. – 1 – 5,1
The direction ratios of AC are 3 – 2, 8 – 3, –11 + 4 i.e. 1, 5, – 7
∴ direction ratios of AB and AC are proportional
∴ AB is parallel to AC.
But A is a common point on both the lines.
∴ A, B, C are collinear points.

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20.

Find the length of the projection of the line segment joining the points P (3, -1, 2) and Q(2, 4, -1) on the line with direction ratios -1, 2, -2.

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