NCERT Solutions class 12 Maths Exercise 11.2

NCERT Solutions class 12 Maths Exercise 11.2 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

Download NCERT solutions for Three Dimensional Geometry as PDF.

NCERT Solutions class 12 Maths Three Dimensional Geometry 

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

Ans. Given: Direction cosines of three lines are

  

For first two lines,

=  =

Therefore, the first two lines are perpendicular to each other.

For second and third lines,

=  =

Therefore, second and third lines are also perpendicular to each other.

For First and third lines,

=  =

Therefore, first and third lines are also perpendicular to each other.

Hence, given three lines are mutually perpendicular to each other.

---

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

Ans. We know that direction ratios of the line joining the points A and B are

 

Again, direction ratios of the line joining the points C (0, 3, 2) and D (3, 5, 6) are

 (say)

For lines AB and CD, =  = 6 + 10 – 16 = 0

Therefore, line AB is perpendicular to line CD.

---

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

Ans.  We know that direction ratios of the line joining the points A (4, 7, 8) and B (2, 3, 4) are

 

 =  (say)

Again direction ratios of the line joining the points C and D (1, 2, 5) are

 =  (say)

For the lines AB and CD,

 

Therefore, line AB is parallel to line CD.

---

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

Ans.  A point on the required line is A (1, 2, 3) =

 Position vector of a point on the required line is

The required line is parallel to the vector

 direction ratios of the required line are coefficient of  in  are

Vector equation of the required line is

 

Where  is a real number.

Cartesian equation of this equation is

---

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  

Ans. Position vector of a point on the required line is  =

The required line is in the direction of the vector is

 Direction ratios of required line are coefficients of  in  =  

 Equation of the required line in vector form is

Where  is a real number.

Cartesian equation of this equation is

---

6.Find the Cartesian equation of the line which passes through the point  and parallel to the line given by  

Ans. Given: A point on the line is  

Equation of the given line in Cartesian form is

 Direction ratios of the given line are its denominators 3, 5, 6

Equation of the required line is

 =

---

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

Ans. Given: The Cartesian equation of the line is =  (say)

General equation for the required line is

Putting the values of  in this equation,

 =

 

---

8.Find the vector and Cartesian equations of the line that passes through the origin and  

Ans.  = Position vector of a point here O (say) on the line = (0, 0, 0) =

 = A vector along the line

=  = Position vector of point A – Position vector of point O

=

 Vector equation of the line is

  

NowCartesian equation of the line

Direction ratios of line OA are

And a point on the line is O (0, 0, 0) =

Cartesian equation of the line =

=  =

Remark:In the solution of the above question we can also take:

 = Position vector of point A =  for vector form and point A as =  for Cartesian form.

Then the equation of the line in vector form is

And equation of line in Cartesian form is

---

9.Find vector and Cartesian equations of the line that passes through the points  and  

Ans. Let  and  be the position vectors of the points A and B respectively.

  and

A vector along the line =  = Position vector of point B – Position vector of point A

  =  =    =

Vector equation of the line is

And another vector equation for the same line is  =

Cartesian equation

Direction ratios of line AB are

Equation of the line is  =

---

10.Find the angle between the following pairs of lines:

(i) and  

(ii) and

Ans. (i)Equation of the first line is

Comparing with ,

 and

(vector  is the position vector of a point on line and  is a vector along the line)

Again, equation of the second line is

Comparing with ,

 and

(vector  is the position vector of a point on line and  is a vector along the line)

Let  be the angle between these two lines, then

 =  =

  

(ii)Comparing the first and second equations with  and  resp.

 and

Let  be the angle between these two lines, then

 =  =

  

---

NCERT Solutions class 12 Maths Exercise 11.2

11.Find the angle between the following pair of lines:

(i) and  

(ii) and  

Ans. (i)Given: Equation of first line is

The direction ratios of this line i.e., a vector along the line is

 =  =

Nowequation of second line is

The direction ratios of this line i.e., a vector along the line is

 =  =

Let  be the angle between these two lines, then

 =  =

 

(ii)Given: Equation of first line is

The direction ratios of this line i.e., a vector along the line is

 =  =

Nowequation of second line is

The direction ratios of this line i.e., a vector along the line is

 =  =

Let  be the angle between these two lines, then

 =  =

 

---

NCERT Solutions class 12 Maths Exercise 11.2

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

Ans. Given: Equation of one line   

 

 Direction ratios of this line are

Again, equation of another line  

 Direction ratios of this line are

Since, these two lines are perpendicular.

Therefore,

 

 

---

NCERT Solutions class 12 Maths Exercise 11.2

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

Ans. Equation of one line

 Direction ratios of this line are  =

Again equation of another line

 Direction ratios of this line are 1, 2, 3 =

Now  =  =

Hence, the given two lines are perpendicular to each other.

---

NCERT Solutions class 12 Maths Exercise 11.2

14.Find the shortest distance between the lines  and  

Ans. Comparing the given equations with  and , we get

 and

Since, the shortest distance between the two skew lines is given by

 ……….(i)

Here,

Putting these values in eq. (i),

Shortest distance

---

NCERT Solutions class 12 Maths Exercise 11.2

15.Find the shortest distance between the lines  and .

Ans. Equation of one line is

Comparing this equation with , we have

Again equation of another line is

Comparing this equation with , we have

   =

Expanding by first row =  =

And

=  =  =

 Length of shortest distance =

=  (numerically)

=

---

NCERT Solutions class 12 Maths Exercise 11.2

16.Find the shortest distance between the lines whose vector equations are

 and

Ans. Equation of the first line is

Comparing this equation with ,

 and

Again equation of second line

Comparing this equation with ,

 and

Now shortest distance  =  ……….(i)

Here

Putting these values in eq. (i),

Shortest distance

---

NCERT Solutions class 12 Maths Exercise 11.2

17.Find the shortest distance between the lines whose vector equations are

 and  

Ans. Equation of first line is

=   =

Comparing this equation with ,

Equation of second line is

=  =

Comparing this equation with ,

Now Shortest distance  =  ……….(i)

Here

Putting these values in eq. (i),

Shortest distance

NCERT Solutions class 12 Maths Exercise 11.2

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

CBSE App for Students

To download NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through the best app for CBSE students and myCBSEguide website.