IB-06three_di_geometry(25-28)

8/9/2019 IB-06three_di_geometry(25-28)

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6. TH REE DIM ENSIONA L GEOMETRY

Synopsis :1. The distance between the points (x 1, y1, z1) and (x 2, y2, z2) is 212212212 )zz()yy()xx( ++ .

2. If A = (x 1, y1, z1), B = (x 2, y2, z2), the co-ordinates of point which divides AB in the ratio l:m

internally is

++

++

++

mlmzlz

,mlmyly

,mlmxlx 121212 .

3. The co-ordinates of the point which divides AB in the ratio l:m externally is

mlmzlz

,mlmyly

,mlmxlx 121212 .

4. Mid point of AB =

+++

2

zz ,

2

yy ,

2

xx 212121 .

5. The centroid of the triangle formed by the points

(x1, y1, z1), (x 2, y2, z2) and (x 3, y3, z3) is

++++++

3zzz

,3

yyy ,

3xxx 321321321 .

6. The centroid of the tetrahedron formed by the points (x 1, y1, z1), (x 2, y2, z2), (x 3, y3, z3) and

(x4, y4, z4) is +++

,4

xxxx 4321 ,4

yyyy 4321 +++ +++

4zzzz 4321 .

7. The line segment joining (x 1, y1, z1) and (x 2, y2, z2) is divided by

i) YZ plane in the ratio x1:x2

ii) ZX plane in the ratio y1:y2

iii) XY plane in the ratio z1:z2

8. If A = (x 1, y1, z1), B = (x 2, y2, z2), P = (x 3, y3, z3) are the vertices of a triangle and BC = a, CA = b,

AB = c, then the incentre of the triangle ABC is

++++

++++

++++

cbaczbzaz

,cbacybyay

,cbacxbxax 321321321 .

9. If a directed line (ray) makes angle , , with positive directions of the axes of x,y and z

respectively the cos , cos , cos are called the direction cosines (d.cs) of the line and these areusually denoted by l,m,n.

10. Direction cosines of i) x axis are (1,0,0) ii) y axis are (0,1,0) iii) z axis are (0,0,1)

11. If (l, m, n) are direction cosines of a ling OP and OP = r then the coordinates p are

(lr, mr, nr).

12. If the coordinates of a point P are (x,y, z) and OP=r then the direction Cosines of OP are (x/r, y/r/

z/r).

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Three Dimensional Geometry

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13. If , , are the angles, made by a line with the positive direction of coordinate axes then cos 2 +cos 2 + cos 2 = 1

14.If , , are the angles, made by a directed line with the positive directions of the coordinate axes,then sin 2 + sin 2 + sin 2 = 2.

15. If (a, b, c) are direction ratio of a line than

++++++ 2222222 cbac

,cba

b,

cba

a are direction cosines of the line.

16. Direction ratios of the line joining the points A (x 1, y1, z1), B (x 2, y2, z2) are (x 2 x 1, y2 y 1, z2 z 1).

17. If l 1, m 1, n1 and l 2, m 2, n2 are real numbers then

(l12 + m 12 + n 12) (l 22 + m 22 + n 22) (l 1l2 + m 1m2 + n 1n2)2 = (m 1n2 m 2n1)2 + (n 1l2 n 2l1) + (l 1m2

l2m2)2

18. If (l 1, m 1,,n1 ) and (l 2,,m2,n2) are direction cosines of two lines and is an angle between then, thencos = l1l2 + m 1m2 + n 1n2.

19. If a line makes equal angles with co-ordinate axes, then d.c.s of a vector are3

1 ,

31

,3

1 and

number of such lines passing through the origin is 4.

20. If (a 1, b 1, c 1) and (a 2, b 2, c 2) are d.c.s of two lines and is the angle between the lines, then cos

=2

22

22

22

12

12

1

212121

cbacba

ccbbaa

++++++ .

21. If two lines are perpendicular, then a 1a2 + b 1 b2 + c 1c2 = 0.

22. If l, m, n be the direction cosines of a line and P = (x 1, y1, z1), Q = (x 2, y2, z2). The projection of PQ

on a line is )zz(n)yy(m)xx(l 121212 ++ .

23. If P = (x 1, y1, z1), Q = (x 2, y2, z2), then

i) The projection of PQ on x-axis is |x 2 x1|.

ii) The projection of PQ on y-axis is |y 2 y1|.

iii) The projection of PQ on z-axis is |z 2 z1|.

24. If l 1, m 1, n1 and l 2, m 2, n2 are d.R.s of a two lines, then the d.R.s of a line perpendicular to these

two lines are m 1n2 m2n1, n1l2 n2l1, l1m2-l2m1.

25. If (l 1, m 1, n 1), (l 2, m 2, n2), (l 3, m 3, n 3) are d.c.s of three mutually perpendicular lines, then d.R.s of

the lines which makes equal angles with the three lines are l 1 + l 2 + l 3, m 1 + m 2 + m 3, n 1 + n 2 + n 3

and each angle is Cos 1(1/ 3 ).




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Three Dimensional Geometry

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26. Angle between any two diagonals of a cube is Cos 1(1/3).

27. Angle between one diagonal of a cube and a diagonal of one face is Cos -1( 3 / 2 ).

28. If l, m, n are d.c.s of a line then the maximum value of lmn is 1/3 3 .29. If , , , are angles made by a line with diagonals of a cube, then cos 2 + cos 2 + cos 2 + cos 2

= 4/3.



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IB-06three_di_geometry(25-28)