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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)

96 Views

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

191 Views

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).

Let A(3, 5, – 4), B(–1, 1, 2), C(–5, –5, –2) be the vertices of ΔABC.
Direction ratios of AB are – 1 – 3, 1 – 5, 2 + 4 i.e. – 4, – 4, 6
Dividing each by

square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus left parenthesis 6 right parenthesis squared end root

equals space square root of 16 plus 16 plus 36 end root space equals space square root of 68 comma space we space get space the space direction space

cosines of the line AB as

negative fraction numerator 4 over denominator square root of 68 end fraction comma space minus fraction numerator 4 over denominator square root of 68 end fraction comma space fraction numerator 6 over denominator square root of 68 end fraction

i.e.

negative fraction numerator 1 over denominator square root of 17 end fraction comma space space space minus fraction numerator 2 over denominator square root of 17 end fraction comma space fraction numerator 3 over denominator square root of 17 end fraction.

Direction ratios of BC are – 5 + 1, –5 –1, –2 –2 i.e. – 4, –6, –4.
Dividing each by

square root of left parenthesis negative 4 right parenthesis squared plus left parenthesis negative 6 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared end root space equals space square root of 16 plus 36 plus 16 end root space equals space square root of 68 comma space we space get space the space

direction ratios of the line BC as

negative fraction numerator 4 over denominator square root of 68 end fraction comma space minus fraction numerator 6 over denominator square root of 68 end fraction comma space minus fraction numerator 4 over denominator square root of 68 end fraction space space space or space space space minus fraction numerator 2 over denominator square root of 17 end fraction comma space minus fraction numerator 3 over denominator square root of 17 end fraction comma space minus fraction numerator 2 over denominator square root of 17 end fraction.

Direction ratios of CA are 3+5, 5+5, -4+2 i.e., 8, 10 -2.

Dividing each by

square root of left parenthesis 8 right parenthesis squared plus left parenthesis 10 right parenthesis squared plus left parenthesis negative 2 right parenthesis squared end root space equals space square root of 64 plus 100 plus 4 end root space equals space square root of 168 comma space we space get space the space

direction ratios of the line CA as

fraction numerator 8 over denominator square root of 168 end fraction comma space fraction numerator 10 over denominator square root of 168 end fraction comma space space minus fraction numerator 2 over denominator square root of 168 end fraction comma space straight i. straight e. space fraction numerator 4 over denominator square root of 42 end fraction comma space fraction numerator 5 over denominator square root of 42 end fraction comma space minus fraction numerator 1 over denominator square root of 42 end fraction.

145 Views

Short Answer Type

14.

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

93 Views

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?

481 Views

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

156 Views

17.

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

119 Views

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

91 Views

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

156 Views

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.

93 Views