Applications of Vectors
Vector Equation of a Straight Line :
Following are the two most useful
forms of the equation of a line.
(i) Line passing through a given point A
and parallel to a vector 
where
is the p.v. of any general point P on
the line and l is any real number.
The vector equation
of a straight line passing through the origin and parallel to a vector
is
=
n
.

(ii) Line passing through two given points
A
and B

For each particular value of l, we get a particular point on the
line. .
Each of the above equations can be written
easily in Cartesian form also.
For example, in case
(i), writing

,
we get 
&⇒
x = a1 + lb1, y = a2
+ lb2, z = a3
+ lb3.
Example.1
What is the vector equation of the line
through the points with position vectors and
..
Solution
Here the line is
passing through the point with p.v.
. At the same time
the line is along the vector
i.e. along the
vector
. Hence the required equation of the
line is 
Example.2
Prove that the equation of the straight line
through the point with p.v. and equally
inclined to vectors is ..
Solution:
Let the required
line be
, where
is
a vector parallel to line. Now 
&⇒
=
0 = 
Hence
is perpendicular to
and
.
&⇒
is
parallel to
= 
&⇒ 
Hence
+
.
Example.3
ABC is a triangle. A line is drawn parallel
to BC to meet AB and AC in D and E respectively. Prove that the median through
A bisects DE..
Solution
Take the vertex A of the triangle ABC as the
origin. Let
be the p.v. of B and C. The mid point
of BC has the p.v. = 
The equation of the median is
.

Let D divide AB in the ratio 1 : m
&⇒
p.v. of 
So E divide AC in the ratio 1: m
&⇒
p.v. of E = 
p.v. of the mid-point of DE =
which lies on the median.
Hence the median bisects DE.
Angle between two lines
If
be the vector equation of two lines in
space and q be the angle between
them, then cos q =
Lines are perpendicular if b1.b2
= 0.
Lines are parallel if b1= lb2
The internal bisector of angle
between unit vectors
and
is
along the vector
+
.
The external bisector is along
-
.
Equation of internal and external bisectors
of the lines
(intersecting at
) are given by
.
Shortest Distance between
two lines:
Two lines in space
can be parallel, intersecting or neither (called skew lines). Let
be two lines.
(i) They intersect if
.
(ii) They are parallel if
are collinear. Parallel lines are of
the form
Perpendicular distance between them
is constant and is equal to
.
(iii) For skew
lines, shortest distance between them (along common perpendicular) is given by
.
Equation of a plane in
vector form:
Following are the four useful ways of
specifying a plane.
(i) A plane at a perpendicular distance d
from the origin and normal to a given direction
has
the equation 
or
(
is a unit vector).

(ii) A plane passing
through the point A
and normal to
has the equation
.

(iii) Parameteric equation of the plane
passing through A
and parallel to the plane of
vectors
is given by 
&⇒.
(iv) Parameteric equation of the plane
passing through points A
, B
C
(A,
B, C non-collinear) is given by

&⇒
.
In Cartesian form, the equation of the plane
assumes the form Ax + By + Cz = D. The vector normal to this plane is
and the perpendicular distance of the
plane from the origin is
.
Perpendicular distance of the plane
from point with position vector
is 
Angle between a line and a
plane
The angle between a
line and a plane is the complement of the angle between the line and the normal
to the plane.
Angle between two planes
The angle between two planes is equal to the angle
between their normal unit vectors
. i.e. cosq = 
Some Miscellaneous Results:
(i) Volume of the tetrahedron ABCD=

(ii) Area of the quadrilateral with
diagonals
=
.

(iii) Reciprocal system of vectors: If
are three non-coplanar vectors, then
a system of vectors
defined by

is called the reciprocal system of vectors
because
.
Further 

The scalar product
of any vector of one system with a vector of other system which does not
correspond to it is zero i.e. 
If
is a reciprocal system to
then
is
also reciprocal system to
.
Example.1
Show that if and are
two non-zero vectors, then and make equal angles with if
Solution
It is evident that
is along the bisector of the angle
between the vectors
and 
&⇒ 
If we take l = 
Then 