Chapter 11. Three Dimensional Geometry

Class 12 Maths Chapter 11Three Dimensional Geometry Exercise 11.1 Questions and Solutions :

1. If a line makes angles 90° , 135° , 45° with the and  axes respectively , find its direction cosines .

Solution:  Let the direction cosines be   .

Here,

We have, ,

and

Therefore, the direction cosine of the line 0 ,  and  .

2. Find the direction cosines of a line which makes equal angles with the coordinate axes .

Solution:  Let the direction cosines be  .

We have,

Therefore, the direction cosine of the line ,  and   or    , and .

3. If a line has the direction ratios – 18 , 12 , – 4 , then what are its direction cosines ?

Solution:  Given, the direction ratios are  – 18 , 12 , – 4 .

Here,

The direction cosines of the line are :

and  

Therefore, the direction cosines are :

4. Show that the points (2 , 3 , 4) , (– 1 , – 2 , 1) , (5 , 8 , 7) are collinear .

Solution:  We know that the direction ratios of the line segment joining  and  are given by  .

Let the points are A(2 , 3 , 4) ,B (– 1 , – 2 , 1) and C (5 , 8 , 7) respectively .

The direction ratios of line joining A and B are :

       – 1 – 2 , – 2 – 3 , 1 – 4  i.e.,  – 3 , – 5 , – 3     

The direction ratios of line joining B and C are :

 5 – (– 1)  ,  8 – (– 2) , 7 – 1 i.e., 6 , 10 , 6

So, the direction ratios of AB and BC are proportional.

 Hence, AB is parallel to BC. But point B is common to both AB and BC.

Therefore, A, B, C are collinear points.

5. Find the direction cosines of the sides of the triangle whose vertices are (3 , 5 , – 4) , (– 1 , 1 , 2) and ( – 5 , – 5 , – 2) .

Solution:  Let the vertices are A(3 , 5 , – 4) , B(– 1 , 1 , 2) and C( – 5 , – 5 , – 2) of a triangle ABC respectively .

[ The direction ratios of the line are :  ]

 The direction ratios of line joining A and B are :

      – 1 – 3 , 1 – 5 , 2 – (– 4)  i.e.,  – 4 , – 4 , 6

The magnitude of AB

 

Therefore, the direction cosines of the side AB are :  

i.e.,  

The direction ratios of line joining B and C are :

      – 5 – (– 1) , – 5 – 1  , – 2 – 2  i.e.,  – 4 , – 6 , – 4  

The magnitude of BC

 

Therefore, the direction cosines of the side BC are :  

i.e.,    

The direction ratios of line joining C and A are :

 3 – (– 5) , 5 – (–5) , – 4 – (– 2)  i.e.,   8 ,  10 , – 2 

The magnitude of AC

 

Therefore, the direction cosines of the side CA are : 

i.e.,  

Class 12 Maths Chapter 11Three Dimensional Geometry Exercise 11.2 Questions and Solutions :

1. Show that the three lines with direction cosines    ;   ;   are mutually perpendicular.

Solution:  We know that the two lines with direction ratios  and  then  .

The direction cosine of the line are  and  .

The direction cosine of the line are and  .

The direction cosine of the line are   and  .

Therefore, the three lines with direction cosines ; ;   are mutually perpendicular.

2. Show that the line through the points (1, – 1, 2), (3, 4, – 2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Solution:  Given, the points are A(1, – 1 , 2) , B(3 , 4 , – 2) , C(0 , 3 , 2) and D(3 , 5 , 6) respectively .

So, the direction ratio of AB are : , ,  

So, the direction ratio of CD are : , ,  

We have ,

Therefore, the points (1, – 1, 2), (3, 4, – 2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

3. Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (– 1, – 2, 1), (1, 2, 5).

Solution: Given, the points are A(4, 7, 8), B(2, 3, 4), C(– 1, – 2, 1)and D (1, 2, 5) respectively .

So, the direction ratio of AB are :, ,   

So, the direction ratio of CD are :  , ,  

We have ,    ,    and 

Therefore, the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (– 1, – 2, 1), (1, 2, 5).

4. Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector  .

Solution:  Since, the line passes through the point (1, 2, 3) , then

And

We have ,

5. Find the equation of the line in vector and in cartesian form that passes through the point with position vector  and is in the direction  .

Solution:  Given,   and   

So, the vector equation of the line is given by 

Let   

Comparing the coefficients of  ,  and  , we get

,  ,

, ,

  ,  , 

Eliminating the parameter  , we get

6. Find the cartesian equation of the line which passes through the point (– 2, 4, – 5) and parallel to the line given by 

Solution: Given, the line passes through the point (– 2, 4, – 5) and parallel to the line given by

The direction ratios of the line are (3, 5 ,6) .
Since, the line passes through the point (– 2, 4, – 5) , the line is

i.e.,  

7. The cartesian equation of a line is  . Write its vector form.

Solution:  Given, the Cartesian equation of a line is

Comparing the given equation with the standard form

Here, ,

Thus , the required line passes through the point ( 5,– 4 , 6) and is parallel to the vector .

Let be the position vector of any point on the line, then the vector equation of the line is given by

 

8. Find the angle between the following pairs of lines:
(i)    and   
(ii)    and    

Solution:  (i) We have,   and   

Here, ,

The angle between the two lines is given by

 

 
(ii) We have,   and   

Here,  and

The angle between the two lines is given by

9. Find the angle between the following pair of lines:
(i)  and

(ii)  and

Solution:  (i) The direction ratios of the line are (2 , 5 ,– 3) .

Here,

and the direction ratios of the line are (– 1 , 8 , 4)

Here,

The angle between the two lines is given by

(ii) The direction ratios of the line are (2 , 2 , 1)

Here,

And the direction ratios of the line are (4 , 1, 8)

Here,

The angle between the two lines is given by

 

 

10. Find the values of p so that the lines and are at right angles.

Solution: Given, the lines

 

 

The direction ratios of the line are

Here,

And the line

The direction ratios of the line are

Here,

We have,

11. Show that the lines   and are perpendicular to each other.

Solution:  Given, the lines’

The direction ratios of the line are (7 , – 5 , 1)

Here,

And the line is

The direction ratios of the line are (1 , 2 , 3) .

Here,

We have,

Therefore, the lines and are perpendicular to each other.

12. Find the shortest distance between the lines
   and   
Solution:  Given, the lines

   

Here,and

and    

Here, and

Hence, the required shortest distance is

units

13. Find the shortest distance between the lines and
Solution:  The lines

i.e.,

Here, ,

And the lines

Here, ,

The shortest distance between the lines

uints

14. Find the shortest distance between the lines whose vector equations are   and   
Solution: We have,  

Here , ,

Again,    

Here,  ,

Hence, the shortest distance between the given lines is given by

units

15. Find the shortest distance between the lines whose vector equations are
   and

Solution:  We have,  

Here,   ,

and

 

Here,  ,

Hence, the shortest distance between the given lines is given by

Miscellaneous Exercise on Chapter 11

1. Find the angle between the lines whose direction ratios are  and  .

Solution:  Here, ,

The angle between the two lines is given by

 

Therefore, the angle between the lines is 90° .

2. Find the equation of a line parallel to x-axis and passing through the origin.

Solution:  let , P be the point on the X-axis  and also the coordinate of the point P is  . 

The coordinate of origin is (0,0,0) .

The direction ratios of OP is

A line parallel to the x-axis has a direction ratio of (1, 0, 0), as it does not have any component along the y-axis or z-axis.

We know that ,the equation of a line passing through the point  with direction ratios  is

So, the equation of a line passing through the point  with direction ratios  is

i,e.,
3. If the lines and are perpendicular, find the value of  .

Solution:  Here, ,

We have,

4. Find the shortest distance between lines   and .

Solution:  We have,  

Here,  ,

and   

Here,   and  

Hence, the shortest distance between the given lines is given by units

5. Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to the two lines:    and  .

Solution: Let the Cartesian equation of the line is 

Since the line passing through the point (1, 2, – 4) , then

The direction ratios of the line are  .

Given the lines

The direction ratios of the line are (3 , – 16 , 7)

Again , the line

The direction ratios of the line are (3 , 8 , – 5) .

From the line (i) and (ii) , we get

From  the line (i) and (iii) , we get

Using cross-multiplication method, we have

Let  

 

The position vector of the point (1 , 2 , – 4) is  .

Therefore, the vector equation of the line is