Definitions and basic Theorems

1. Analytical Geometry of Three Dimensions
18UMTC22
I B.Sc (Maths)
T.KALAISELVI M.Sc., M.Phil
Department of Mathematics (SF)

2. Definition
A plane in 𝑅3
is defined to be
the locus of a point (x, y, z)
satisfying a linear equation of
the form ax + by + cz + d = 0
where a, b, c are not all zero.

3. Theorem 1
Equation of a plane passing through a
given point (𝑥1, 𝑦1, 𝑧1) and having a
normal whose d.r are a, b,c is given by
a(𝑥 − 𝑥1) + b (𝑦 − 𝑦1) + c (𝑧 − 𝑧1) =
0.

4. Definition
Angle between two planes
is defined to be the angle
between the normal to
them from any point.

5. Theorem 2
The angle 𝜃 between the planes
ax + by + cz + d = 0 and
𝑎1 𝑥 + 𝑏1 𝑦 + 𝑐1 𝑧 + 𝑑1 = 0 is given by
cos 𝜃 = ±
𝑎𝑎1 + 𝑏𝑏1 + 𝑐𝑐1
𝑎2 𝑎1
2

6. Definition
The equation of a line passing
through 𝑥1, 𝑦1, 𝑧1 and having
direction rations (a, b, c) are
given by
𝑥 − 𝑥1
𝑎
=
𝑦 − 𝑦1
𝑏
=
𝑧 − 𝑧1
𝑐

7. Definition
If A(𝑥1, 𝑦1, 𝑧1) and B(𝑥2, 𝑦2, 𝑧2) are
two points on a line then the direction
rations of the line are 𝑥2 − 𝑥1, 𝑦2 −
𝑦1 , 𝑧2 − 𝑧1.
Therefore the equation of the line is
𝑥 − 𝑥1
𝑥2 − 𝑥1
=
𝑦 − 𝑦1
𝑦2 − 𝑦1
=
𝑧 −𝑧1
𝑧2 − 𝑧1
.

8. Theorem 3
The condition for two lines
𝑥 − 𝑥1
𝑙1
=
𝑦 − 𝑦1
𝑚1
=
𝑧 − 𝑧1
𝑛1
and
𝑥 − 𝑥2
𝑙2
=
𝑦 − 𝑦2
𝑚2
=
𝑧 − 𝑧2
𝑛2
to be coplanar is
𝑥2 − 𝑥1 𝑦2 − 𝑦1 𝑧2 − 𝑧1
𝑙1 𝑚1 𝑛1
𝑙2 𝑚2 𝑛2
= 0

9. Theorem 4
The angle between the line
𝑥 − 𝑥1
𝑙
=
𝑦 − 𝑦1
𝑚
=
𝑧 − 𝑧1
𝑛
and the line
𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 + 𝑑 = 0 is given by
sin 𝜃 =
𝑎𝑙 + 𝑏𝑚 + 𝑐𝑛
𝑎2 + 𝑏2 + 𝑐2 𝑙2 + 𝑚2 + 𝑛2

10. Definition
Two straight lines in space which are
not coplanar are called Skew lines
Note
There is only one straight line which is
perpendicular to both the skew lines.