NCERT Solutions class 12 Maths Exercise 10.4



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NCERT Solutions class 12 Maths Exercise 10.4 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 Vector Algebra as PDF.

NCERT Solutions class 12 Maths Exercise 10.4

NCERT Solutions class 12 Maths Vector Algebra

Vector Algebra

Exercise 10.4

1. Find  if  and  

Ans. Given:  and

Expanding along first row,

 =

=

 


2. Find a unit vector perpendicular to each of the vectors  and  where  and  

Ans. Given:  and

On Adding   =  +  =

On Subtracting   =    =

Therefore,

Expanding along first row =

 

  =

Therefore, a unit vector perpendicular to both   and  is

 =  =


3. If a unit vector  makes an angle  with   with  and an acute angle  with  then find  and hence, the components of .

Ans. Let  be a unit vector.  ……….(i)

   

Squaring both sides,   ……….(ii)

Given: Angle between vectors  and  is

    

  ……….(iii)

Again, given Angel between vectors  and  is

    

  ……….(iv)

Again, given Angel between vectors  and  is  where  is acute angle.

    

  ……….(v)

Putting the values of  and  in eq. (ii),

   

    

Since  is acute angle, therefore  is positive and hence   

From eq. (v),

Putting values of  and  in eq. (i),

 Components of  are coefficients of  in

  and angle


4. Show that  

Ans. L.H.S. =  =   

=  =  = R.H.S.


5. Find  and  if  

Ans. Given:

 

Expanding along first row,

 =

Comparing the coefficients of  on both sides, we have

  ……….(i)

  ……….(ii)

And   ……….(iii)

From eq. (ii),     

From eq. (iii),     

Putting the values of  and  in eq. (i),

    0 = 0

Therefore,  and


6. Given that  and  What can you conclude about the vectors  and  

Ans. Given:   

   or   or   

   or   or vector  is perpendicular to  …..(i)

Again, given      

   or   or   

     or   or vector  and  are collinear or parallel. …..(ii)

Since, vectors  &  are perpendicular to each other as well as parallel are not possible. ..(iii)

Therefore, from eq. (i), (ii) and (iii),  either   or

  and


7. Let the vectors  be given as  then show that  

Ans. Given: Vector  and

 

Now L.H.S. =

=  +

[By Property of Determinants]

=  = R.H.S.


8. It either  and  then  Is the converse true? Justify your answer with an example.

Ans. Given: Either  or

  or ……….(i)

 

   [Using eq. (i)]

  [By definition of zero vector]

But the converse is not true.

Let   

  is a non-zero vector.

Let

 

  is a non-zero vector.

But

Taking 2 common from R3 =   [ R2 and R3 are identical]


NCERT Solutions class 12 Maths Exercise 10.4

9. Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).

Ans. Vertices of  are A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).

 Position vector of point A = (1, 1, 2) =

Position vector of point B = (2, 3, 5) =

Position vector of point C = (1, 5, 5) =

Now  = Position vector of point B – Position vector of point A

 

=  

=

And  = Position vector of point C – Position vector of point A

 

=  

=

  x  =

=  =

Now Area of triangle ABC =

=  sq. units


NCERT Solutions class 12 Maths Exercise 10.4

10. Find the area of the parallelogram whose adjacent sides are determined by the vectors   and  

Ans. Given: Vectors representing two adjacent sides of a parallelogram are

 and 

  =  =

Now Area of parallelogram =

=  sq. units


NCERT Solutions class 12 Maths Exercise 10.4

11. Let the vectors  and  such that  then  is a unit vector, if the angle between  and  is:

(A)  

(B)  

(C)  

(D)  

Ans. Given:  and  is a unit vector.

    , where  is the angle between  and

 

 

 

 

 

Therefore, option (B) is correct.


NCERT Solutions class 12 Maths Exercise 10.4

12. Area of a rectangle having vertices A, B, C and D with position vectors  and  respectively is:

(A)  

(B) 1

(C) 2

(D) 4

Ans. Given: ABCD is a rectangle.

Now  = Position vector of point B – Position vector of point A

=

=

 AB =

And   = Position vector of point D – Position vector of point A

=

=

 AD =

 Area of rectangle ABCD = Length x Breadth = AB x AD = 2 x 1 = 2 sq. units

Therefore, option (C) is correct.

NCERT Solutions class 12 Maths Exercise 10.4

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