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 = n .

(ii) Line passing through two given points Aand 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 = a_{1} + lb_{1}, y = a_{2}
+ lb_{2}, z = a_{3}
+ lb_{3}.

*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 line is

*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 + .

*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

&⇒
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.

If be the vector equation of two lines in
space and *q** *be the angle between
them, then cos q =

Lines are perpendicular if b_{1}.b_{2}
= 0.

Lines are parallel if b_{1}= lb_{2}

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 .

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 .

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 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

The angle between a line and a plane is the complement of the angle between the line and the normal to the plane.

The angle between two planes is equal to the angle between their normal unit vectors . i.e. cosq =

(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 .

*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 =

Then