# Exercise 28.1

Question 1

Find the vector and cartesian equations of the line through the point and which is parallel to the vector

Sol :

We know that the vector equation of a line passing through a point with position vector and parallel to is

Here,

Vector equation of the required line is given by

…..1

Here, is a parameter

Reducing (1) to cartesian form, we get

Comparing the coefficients of and ,we get

Hence, the cartesian form of (1) is

Question 2

Find the vector equation of the line passing through the points and .

Sol :

We know that the vector equation of a line passing through the points with position vectors  and parallel to is  , where is some scalar.

Here,

Vector equation of the required line is given by

Here is a parameter.

Question 3

Find the vector equation of a line which is parallel to the vector and which passes through the points . Also , reduce it to cartesian form.

Sol :

We know that the vector equation of a line passing through a point with position vector and parallel to is

Here,

Vector equation of the required line is given by

…..1

Here, is a parameter

Reducing (1) to cartesian form, we get

Comparing the coefficients of and ,we get

Hence, the cartesian form of (1) is

Question 4

A line passes through the point with position vector  and is in the direction of  .  Find equations of the line in vector and cartesian form.

Sol :

We know that the vector equation of a line passing through a point with position vector and parallel to is

Here,

Vector equation of the required line is given by

…..1

Here, is a parameter

Reducing (1) to cartesian form, we get

Comparing the coefficients of and ,we get

Hence, the cartesian form of (1) is

Question 5

ABCD is a parallelogram. The position vectors of the points A,B,C are respectively and  . Find the vector equation of the line BD. Also, reduce it to cartesian form.

Sol :

We know that the position vector of the mid-point of  and  is .

Let the position vector of point D be

Position vector of mid-point of A and C = Position vector of mid-point of B and D

Comparing the coefficients of and ,we get

Position vector of points

The vector equation of line BD passing through the points with position vectors and is

Here,

Vector equation of the required line is given by

…..1

Here, is a parameter

Reducing (1) to cartesian form, we get

Comparing the coefficients of and ,we get

Hence, the cartesian form of (1) is

Question 6

Find in vector form as well as in cartesian form, the equation of the line passing through the points A and B

Sol :

We know that the vector equation of a line passing through a point with position vector and parallel to is

Here,

Vector equation of the required line is given by

…..1

Here, is a parameter

Reducing (1) to cartesian form, we get

Comparing the coefficients of and ,we get

Hence, the cartesian form of (1) is

Question 7

Find in vector form as well as in cartesian form, the equation of the line passing through the points and parallel to the vector . Reduce the corresponding equation in cartesian form.

Sol :

We know that the vector equation of a line passing through a point with position vector and parallel to is

Here,

Vector equation of the required line is given by

…..1

Here, is a parameter

Reducing (1) to cartesian form, we get

Comparing the coefficients of and ,we get

Hence, the cartesian form of (1) is

Question 8

Find the vector equation of a line passing through and parallel to the line whose equations are

Sol :

We know that the vector equation of a line passing through a point with position vector and parallel to is

Here,

Vector equation of the required line is given by

…..1

Here, is a parameter

Question 9

The cartesian equations of a line are . Find a vector equation for the line.

Sol :

The cartesian equation of the given line is  .

It can be re-written as

Thus, the given line passes through the point having position vector

and is parallel to the vector

We know that the vector equation of a line passing through a point with position vector and parallel to is

Vector equation of the required line is given by

Here, is a parameter

Question 10

Find the cartesian equation of a line passing through and parallel to the line whose equations are . Also, reduce the equation obtained in vector form.

Sol :

We know that the cartesian equation of a line passing through a point with position vector and parallel to the vector is

Here,

Here,

Cartesian equation of the required line is

We know that the cartesian equation of a line passing through a point with position vector and parallel to is

Here, the line is passing through the point and its direction ratios are proportional to 1,2,-2

Vector equation of the required line is

Question 11

Find the direction cosines of the line . Also, reduce it to vector form.

Sol :

The cartesian equation of the given line is

It can be re-written as

This shows that the given line passes through the point and its direction ratios are proportional to -2 , 6 , -3 .

So , the direction ratios are

Thus,the given line passing through a point with position vector and parallel to is

Here,

Vector equation of the required line is

Here, is a parameter.

Question 12

The cartesian equations of a line are x = ay,+ b , z = cy + d .Find its direction ratios and reduce it to vector form.

Sol :

The cartesian equation of the given line is

x = ay + b ,  z = cy + d

It can be re-written as

Thus, the given line passes through the point and its direction ratios are proportional to a,1,c . It is also parallel to the vector

We know that the vector equation of a line passing through a point with position vector and parallel to is

Vector equation of the required line is

Here, is a parameter.

Question 13

Find the vector equation of a line passing through the point with position vector and parallel to the line joining the points with position vector  and and parallel to the vector is

Here,

Vector equation of the required line is

….1

Here,  is a parameter

Reducing (1) to cartesian form , we get

Comparing the coefficients of and ,we get

Hence, the cartesian form of (1) is

Question 14

Find the points on the line at  a distance of 5 units from the point p

Sol :

The coordinates of any point on the line  are given by

….1

Let the coordinates of the desired point be

[Squaring both sides]

Substituting the values of in (1), we get the coordinates of the desired point as and

Question 15

Show that the points whose position vectors are ,$7\hat{i}-\hat{k}$ collinear.

Sol :

Let the given points be P, Q and R and let their position vector be and  respectively.

Vector equation of line passing through P and Q is

…1

If points P,Q and R are collinear , the R must satisfy (1).

Replacing by  in (1) , we get

…1

Comparing the coefficients , and  , we get

These three equations are consistent, i.e. they give the same value of  . Hence , the given three points are collinear

Question 16

Find the cartesian and vector equations of a line which passes through the point and is parallel to the line

Sol :

we have

It can be re-written as

This shows that the given line passes through the point and its direction ratios are proportional to -2 , 14 , 3 .

Thus , the parallel vector is

We know that the vector equation of a line passing through a point with position vector and parallel to the vector is

Here,

Vector equation of the required line is

…..1

Here, is a parameter

Reducing (1) to cartesian form, we get

Comparing the coefficients of and ,we get

Hence, the cartesian form of (1) is

Question 17

The cartesian equation of a line are . Find the fixed point through which it passes , its direction ratios and also its vector equation.

Sol :

The cartesian equation of the given line is

It can be re-written as

Thus , the given line passes through the point and its direction ratios are proportional to . It is parallel to the vector

We know that the vector equation of a line passes through a point with position vector is

Vector equation of the required line is

Here , is a parameter.