Summary of Class 12 Maths Three Dimensional Geometry
Distance Formula
If A(x1,
y1, z1) and B(x2, y2, z2)
are two points in the 3-D space, then the distance between them is given by D =
√[(x2 –
x1)2 + (y2 – y1)2 + (z2 – z1)2].
The distance of
a point P(x, y, z) from the origin O(0, 0, 0) is therefore 
.
Direction
Cosines and Direction Ratios of a Line
Let a line makes
angles 
with the positive directions of x, y
and z-axes, respectively. Then, l = cos 
are called direction cosines of the line. The relation
among l, m and is l2 + m2 + n2
= 1.
Direction
ratios of a line are the numbers which are proportional to the direction
cosines of a line. If l, m and n are the direction cosines
and a, b and c are the direction ratios of a line, then
l = 
, m= 
and n = 
.
Skew Lines
Two straight lines in space which are neither parallel nor
intersecting are called skew lines. Thus, the two lines lie in different
planes.
Angle between skew lines is
the angle between two intersecting lines drawn from any point (preferably
through the origin) parallel to each of the skew lines.
Equation of a Line
(i) Vector equation of a line that
passes through the given point whose position vector is 
and parallel to a
given vector 
is 
= 
+ 
(ii) Equation of a line through a
point (x1, y1, z1) and
having direction cosines l, m, n is
= 
= 
(iii) Equation of a line through a
point (x1, y1, z1) and
having direction ratios a, b, c is
= 
= 
(iv) The
vector equation of a line which passes through two points whose position
vectors
are and is = + 
( - 
.
Angle Between Two Lines
(i) If l1, m1,
n1 and l2, m2, n2
are the direction cosines of two lines; and θ is the acute angle between the
two lines, then
cos θ = | l1l2 + m1m2
+ n1n2|.
(ii) Angle between two lines with equations =
+ 
and =
+ 
is given by 
= 
.
(iii) If a1, b1,
c1 and a2, b2, c2
are the direction ratios of two lines and θ is the acute angle between the two
lines, then
= 
NOTE: If two lines are perpendicular, then 
= 900. Then
= 0.
Shortest Distance Between Two Lines
(i) Shortest distance between two skew lines is the line segment perpendicular to both the lines.
(ii) Shortest distance between two lines whose vector equations are
= + and = + is given by 
.
(iii) Shortest distance between
the parallel lines = + and = + is given by 
.
(iv) Shortest distance between the
lines 
=
= 
and 
=
= 
is
