3D Geometry - 1
3-D GEOMETRY
Introduction
Three dimensional geometry developed accordance to Einsteins field equations. It is useful in several branches of science like it is useful in Electromagnetism. It is used in computer alogorothms to construct 3D models that can be interactively experinced in virtual reality fashion. These models are used for single view metrology. 3-D Geometry as carrier of information about time by Einstein. 3-D Geometry is extensively used in quantum & black hole theory. Section Formula:
(1) Integral division: If R(x, y, z) is point dividing join of P(x1, y1, z1) & Q(x2, y2, z3) in ratio of m : n.
Then, x =



(2) External division: Coordinates of point R which divides join of P(x1, y1, z1) & Q(x2, y2, z2) externally in ratio m : n are

Illustration: Show that plane ax + by + cz + d = 0 divides line joining (x1, y1, z1) & (x2, y2, z2) in ratio of

Ans: Let plane ax + by + cz + d = 0 divides line joining (x1, y1, z1) & (x2, y2, z2) in ratio K : 1



must satisfy eq. of plane.
ax + by + cz + d = 0
[Dumb Question: Why coordinates of P satisfy eq. of plane ?
Ans: Point P lies in the plane so, it satisfy eq. of plane.]





Direction Cosines:

Let












& lies 0






Note: (i) Direction cosines of x-axis are (1, 0, 0)
Direction cosines of y-axis are (0, 1, 0)
Direction cosines of z-axis are (0, 0, 1)
(ii) Suppose OP be any line through origin O which has direction l, m, n
(r cos







or x = lr, y = mr, z = nr
(iii) l2 + m2 + n2 = 1
Proof: |



|





(iv)

Direction ratios: Suppose l, m, & n are direction cosines of vector


Suppose a, b, c are direction ratios of vector








a2






l = ±



Note: (i) If




(ii) Direction ratios of line joining two given points
(x1, y1, z1) & (x2, y2, z2) is (x2 - x1, y2 - y1, z2 - z1)
(iii) If direction ratio’s of



(iv) Projection of segment joining points P(x1, y1, z1) and Q(x2, y2, z2) on a line with direction cosines l, m, n, is:
(x2 - x1)l + (y2 - y1)m + (z2 - z1)n
Illustration: If a line makes angles






Ans: Line is making



Then, direction cosines are l = cos



But l2 + m2 + n2 = 1
cos2



1 - sin2







Angle b/w two vectors in terms of direction cosines & direction ratios:
(i) Suppose




Dumb Question: How

Ans:

But l12 + m12 + n12 = 1

So,

cos


cos![]() |
(ii) IF a1, b1, c1 and a2, b2, c2 are d.r.s of




cos


cos


Note: (i) If two lines are

cos

or a1a2 + b1b2 + c1c2 = 0
(ii) If two lines are || then
cos


or
