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 ,
[latex]7\hat{i}-\hat{k}[/latex] 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.