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 = 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 = 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 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 .