IntroductionThree 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.
(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 = , y = , z =
(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
Coordinates of P
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.]
a(Kx2 + x1) + b(Ky2 + y1) + c(Kz2 + z1) + d(K + 1) = 0
K(ax2 + by2 + cz2 + d) + (ax1 + by1 + cz1 + d) = 0
K = -
Let is a vector , , inclination with x, y & z-axis respectively. Then cos, cos & cos are direction cosines of . They denoted by
, , direction angles.
& 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, r cos, r cos) where OP = r
coordinates of P are (r cos, r cos, r cos)
or x = lr, y = mr, z = nr
(iii) l2 + m2 + n2 = 1
Proof: || = || =
||2 = x2 + y2 + z2 = l2||2 + m2||2 + n2||2
l2 + m2 + n2 = 1
Direction ratios: Suppose l, m, & n are direction cosines of vector & a, b, c are no.s such that a, b, c are proportional to l, m, n. These a, b, c are c/d direction ratios.
Suppose a, b, c are direction ratios of vector having direction cosines l, m, n. Then,
= l = a, m = b, n = c
l2 + m2 + n2 = 1
a22 + b22 + c22 = 1 = ±
l = ± , m = ± , n = ±
Note: (i) If having direction cosines l, m, n. Then, l = , m = , n =
(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 are a, b, c
(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 , & with coordinate axes, prove that sin2 + sin2 + sin2 = 2
Ans: Line is making , & with coordinate axes.
Then, direction cosines are l = cos, m = cos & n = cos.
But l2 + m2 + n2 = 1
cos2 + cos2 + cos2 = 1
1 - sin2 + 1 - sin2 + 1 - cos2 = 1
sin2 + sin2 + sin2 = 2
Angle b/w two vectors in terms of direction cosines & direction ratios:
(i) Suppose & are two vectors having d.c’s l1, m1, n1 & l2, m2, n2 respectively. Then,
Dumb Question: How ?
But l12 + m12 + n12 = 1
|cos = l1l2 + m1m2 + n1n2|
(ii) IF a1, b1, c1 and a2, b2, c2 are d.r.s of & . Then
Note: (i) If two lines are then,
cos = 0 or l1l2 + m1m2 + c1c2 = 0
or a1a2 + b1b2 + c1c2 = 0
(ii) If two lines are || then
cos = 1 or