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 Aand 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 =
(ii) Line passing through two given points
For each particular value of l, we get a particular point on the
Each of the above equations can be written
easily in Cartesian form also.
For example, in case
x = a1 + lb1, y = a2
+ lb2, z = a3
What is the vector equation of the line
through the points with position vectors and
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
Prove that the equation of the straight line
through the point with p.v. and equally
inclined to vectors is ..
Let the required
line be , where is
a vector parallel to line. Now
Hence is perpendicular to and .
parallel to =
Hence + .
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..
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
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
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 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
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
or (is a unit vector).
(ii) A plane passing
through the point Aand normal to has the equation .
(iii) Parameteric equation of the plane
passing through Aand parallel to the plane of
vectors is given by
(iv) Parameteric equation of the plane
passing through points A, BC(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
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
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 .
Show that if and are
two non-zero vectors, then and make equal angles with if
It is evident that is along the bisector of the angle
between the vectors and
If we take l =