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RD Sharma Solution for Class 12 Science Mathematics Chapter 24 - Scalar Or Dot Product

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Our RD Sharma Textbook Solutions are considered extremely helpful for solving the tough questions which are asked in the CBSE Class 12 exam. TopperLearning Textbook Solutions are compiled by subject experts. Herein, you can find all the answers to the textbook questions for Chapter 24 - Scalar Or Dot Product.

All our solutions are created in accordance with the latest CBSE syllabus, and they are amended from time to time to provide the most relevant answers. Our free RD Sharma Solutions for CBSE Class 12 Mathematics will strengthen your fundamentals in Mathematics and will help you in your attempts to score more marks in the final examination. CBSE Class 12 students can refer to our solutions any time — while doing their homework and while preparing for the exam.

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RD Sharma Solution for Class 12 Science Mathematics Chapter 24 - Scalar Or Dot Product Page/Excercise 24.1

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5 (i)


Solution 5 (ii)

Solution 5 (iii)

Solution 5 (iv)

Solution 5 (v)

Solution 6

Solution 7(i)

Solution 7(ii)

begin mathsize 12px style table attributes columnalign left end attributes row cell Let text    end text the text    end text unknown text    end text vector text    end text be text    end text apostrophe straight a with rightwards arrow on top equals straight a subscript 1 straight i with hat on top plus straight b subscript 1 straight j with hat on top plus straight c subscript 1 straight k with hat on top apostrophe end cell row cell straight b with rightwards arrow on top equals straight i with hat on top minus straight j with hat on top plus straight k with hat on top comma text    end text straight c with rightwards arrow on top equals 2 straight i with hat on top plus straight j with hat on top minus 3 straight k with hat on top comma straight d with rightwards arrow on top equals straight i with hat on top plus straight j with hat on top plus straight k with hat on top end cell row cell It text    end text is text    end text given text    end text that text    end text straight a with rightwards arrow on top. straight b with rightwards arrow on top equals 4 end cell row cell straight a subscript 1 minus straight b subscript 1 plus straight c subscript 1 equals 4..... left parenthesis straight i right parenthesis end cell row cell straight a with rightwards arrow on top. straight c with rightwards arrow on top equals 0 end cell row cell 2 straight a subscript 1 plus straight b subscript 1 minus 3 straight c subscript 1 equals 0....... left parenthesis ii right parenthesis end cell row cell straight a with rightwards arrow on top. straight d with rightwards arrow on top equals 2 end cell row cell straight a subscript 1 plus straight b subscript 1 plus straight c subscript 1 equals 2........ left parenthesis iii right parenthesis end cell row blank row cell Solving text    end text left parenthesis straight i right parenthesis comma left parenthesis ii right parenthesis text    end text and text    end text left parenthesis iii right parenthesis comma end cell row cell straight a subscript 1 equals 2 comma straight b subscript 1 equals negative 1 comma straight c subscript 1 equals 1 end cell row blank row cell therefore the text    end text vector text    end text straight a with rightwards arrow on top equals 2 straight i with hat on top minus straight j with hat on top plus straight k with hat on top end cell end table end style

Solution 8 (i)

Solution 8 (ii)

Solution 9

Solution 10

Solution 11


Solution 12


Solution 13


Solution 14

Solution 15

  

Solution 16

begin mathsize 12px style table attributes columnalign left end attributes row cell straight p with rightwards arrow on top equals 5 straight i with hat on top plus straight lambda straight j with hat on top minus 3 straight k with hat on top text   end text and text   end text straight q with rightwards arrow on top text   end text equals straight i with hat on top plus 3 straight j with hat on top minus 5 straight k with hat on top end cell row cell straight p with rightwards arrow on top plus straight q with rightwards arrow on top text   end text end cell row cell equals 5 straight i with hat on top plus straight lambda straight j with hat on top minus 3 straight k with hat on top plus straight i with hat on top plus 3 straight j with hat on top minus 5 straight k with hat on top end cell row cell equals 6 straight i with hat on top plus left parenthesis straight lambda plus 3 right parenthesis straight j with hat on top minus 8 straight k with hat on top end cell row blank row cell straight p with rightwards arrow on top minus straight q with rightwards arrow on top text   end text end cell row cell equals 5 straight i with hat on top plus straight lambda straight j with hat on top minus 3 straight k with hat on top minus straight i with hat on top minus 3 straight j with hat on top plus 5 straight k with hat on top end cell row cell equals 4 straight i with hat on top plus left parenthesis straight lambda minus 3 right parenthesis straight j with hat on top plus 2 straight k with hat on top end cell row blank row cell left parenthesis straight p with rightwards arrow on top plus straight q with rightwards arrow on top right parenthesis. left parenthesis straight p with rightwards arrow on top minus straight q with rightwards arrow on top right parenthesis equals 0 end cell row cell rightwards double arrow left square bracket 6 straight i with hat on top plus left parenthesis straight lambda plus 3 right parenthesis straight j with hat on top minus 8 straight k with hat on top right square bracket. left square bracket 4 straight i with hat on top plus left parenthesis straight lambda minus 3 right parenthesis straight j with hat on top plus 2 straight k with hat on top right square bracket equals 0 end cell row cell rightwards double arrow 24 plus left parenthesis straight lambda squared minus 9 right parenthesis minus 16 equals 0 end cell row cell rightwards double arrow straight lambda squared minus 9 plus 8 equals 0 end cell row cell rightwards double arrow straight lambda squared minus 1 equals 0 end cell row cell therefore straight lambda equals plus-or-minus 1 end cell end table end style

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24


Solution 25


Solution 26

Solution 27

Solution 28

Solution 29


Solution 30(i)

Solution 30(ii)

Solution 31(i)

Solution 31(ii)

Solution 31(iii)

Solution 32(i)

Solution 32(ii)

Solution 32(iii)

Solution 33(i)

Solution 33(ii)

Solution 34

 

Solution 35

Solution 36

Solution 37

Solution 38

Solution 39

Solution 40

Solution 41

Solution 42

Solution 43

Solution 44

Solution 45

Solution 46

Solution 47

Solution 48

Solution 49

begin mathsize 12px style table attributes columnalign left end attributes row cell vertical line straight a with rightwards arrow on top plus straight b with rightwards arrow on top vertical line squared equals vertical line straight b with rightwards arrow on top vertical line squared end cell row cell rightwards double arrow left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis equals straight b with rightwards arrow on top. straight b with rightwards arrow on top end cell row cell rightwards double arrow straight a with rightwards arrow on top. straight a with rightwards arrow on top plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight b with rightwards arrow on top. straight a with rightwards arrow on top plus straight b with rightwards arrow on top. straight b with rightwards arrow on top equals straight b with rightwards arrow on top. straight b with rightwards arrow on top end cell row cell rightwards double arrow straight a with rightwards arrow on top. straight a with rightwards arrow on top plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight b with rightwards arrow on top. straight b with rightwards arrow on top equals straight b with rightwards arrow on top. straight b with rightwards arrow on top end cell row cell rightwards double arrow straight a with rightwards arrow on top. straight a with rightwards arrow on top plus 2 straight a with rightwards arrow on top. straight b with rightwards arrow on top equals 0 end cell row cell rightwards double arrow straight a with rightwards arrow on top. left parenthesis straight a with rightwards arrow on top plus 2 straight b with rightwards arrow on top right parenthesis equals 0 end cell row cell therefore straight a with rightwards arrow on top plus 2 straight b with rightwards arrow on top text    end text is text    end text perpendicular text    end text to text   end text straight a with rightwards arrow on top. end cell end table end style

RD Sharma Solution for Class 12 Science Mathematics Chapter 24 - Scalar Or Dot Product Page/Excercise 24.2

Solution 1

  

Solution 2

 

  

 

 

  

Solution 3

 

  

  

Solution 4

 

  

  

Solution 5

 

  

Solution 6

  

Solution 7

 

  

Solution 8

 

  

  

Solution 9

  

Solution 10

 

  

 

 

  

RD Sharma Solution for Class 12 Science Mathematics Chapter 24 - Scalar Or Dot Product Page/Excercise MCQ

Solution 1

Correct option: (c)

  

Solution 2

Correct option: (b)

  

Solution 3

Correct option: (d)

  

 

Solution 4

Correct option: (c)

  

 

Solution 5

Correct option: (b)

  

Solution 6

Correct option: (c)

  

 

Solution 7

Correct option: (a)

  

Solution 8

Correct option: (b)

  

Solution 9

Correct option: (b)

  

 

Solution 10

Correct option: (c)

  

Solution 11

Correct option: (d)

  

 

Solution 12

Correct option: (b)

  

 

Solution 13

Correct option: (b)

  

Solution 14

Correct option: (c)

  

Solution 15

Correct option: (c)

  

Solution 16

Correct option: (a)

  

 

Solution 17

Correct option: (b)

  

Solution 18

Correct option: (c)

  

 

Solution 19

Correct option: (a)

  

Solution 20

Correct option: (c)

  

Solution 21

Correct option: (c)

 

NOTE: Answer not matching with back answer.

 

 

Solution 22

Correct option: (d)

 

NOTE: Answer not matching with back answer.

Solution 23

Correct option: (d)

  

Solution 24

Correct option: (b)

  

Solution 25

Correct option: (d)

  

 

Solution 26

Correct option: (d)

  

RD Sharma Solution for Class 12 Science Mathematics Chapter 24 - Scalar Or Dot Product Page/Excercise 24VSAQ

Solution 1

Solution 2

Solution 3

Solution 4

Solution 5

Solution 6

Solution 7

Solution 8

Solution 9

Solution 10

Solution 11

Solution 12

Solution 13


Solution 14

Solution 15

Solution 16

Solution 17

Solution 18

Solution 19

Solution 20

Solution 21

Solution 22

Solution 23

Solution 24

Solution 25

Solution 26

Solution 27

Solution 28

Solution 29

Solution 30

Solution 31

Solution 32

Solution 33

begin mathsize 12px style table attributes columnalign left end attributes row cell Let text    end text straight a with rightwards arrow on top equals straight i with hat on top plus 3 straight j with hat on top plus 7 straight k with hat on top end cell row cell straight b with rightwards arrow on top equals 2 straight i with hat on top minus 3 straight j with hat on top plus 6 straight k with hat on top end cell row blank row cell Projection text    end text of text    end text straight a with rightwards arrow on top text    end text on text    end text straight b with rightwards arrow on top end cell row cell equals straight a with rightwards arrow on top. fraction numerator straight b with rightwards arrow on top over denominator vertical line straight b with rightwards arrow on top vertical line end fraction end cell row cell equals left parenthesis straight i with hat on top plus 3 straight j with hat on top plus 7 straight k with hat on top right parenthesis. fraction numerator left parenthesis 2 straight i with hat on top minus 3 straight j with hat on top plus 6 straight k with hat on top right parenthesis over denominator vertical line 2 straight i with hat on top minus 3 straight j with hat on top plus 6 straight k with hat on top vertical line end fraction end cell row cell equals fraction numerator left parenthesis straight i with hat on top plus 3 straight j with hat on top plus 7 straight k with hat on top right parenthesis. left parenthesis 2 straight i with hat on top minus 3 straight j with hat on top plus 6 straight k with hat on top right parenthesis over denominator vertical line 2 straight i with hat on top minus 3 straight j with hat on top plus 6 straight k with hat on top vertical line end fraction end cell row cell equals fraction numerator 1 cross times 2 plus 3 cross times left parenthesis negative 3 right parenthesis plus 7 cross times 6 over denominator vertical line square root of 2 squared plus left parenthesis negative 3 right parenthesis squared plus 6 squared end root vertical line end fraction end cell row cell equals fraction numerator 33 over denominator square root of 49 end fraction end cell row cell equals 33 over 7 end cell end table end style

Solution 34

Solution 35

Solution 36

begin mathsize 12px style table attributes columnalign left end attributes row cell Let text    end text straight a with rightwards arrow on top equals 7 straight i with hat on top plus straight j with hat on top minus 4 straight k with hat on top comma straight b with rightwards arrow on top equals 2 straight i with hat on top plus 6 straight j with hat on top plus 3 straight k with hat on top end cell row cell Projection text    end text of text    end text straight a with rightwards arrow on top text    end text on text    end text straight b with rightwards arrow on top equals straight a with rightwards arrow on top. fraction numerator straight b with rightwards arrow on top over denominator vertical line straight b with rightwards arrow on top vertical line end fraction end cell row cell equals left parenthesis 7 straight i with hat on top plus straight j with hat on top minus 4 straight k with hat on top right parenthesis. fraction numerator left parenthesis 2 straight i with hat on top plus 6 straight j with hat on top plus 3 straight k with hat on top right parenthesis over denominator vertical line square root of 2 squared plus 6 squared plus 3 squared end root vertical line end fraction end cell row cell equals fraction numerator left parenthesis 7 straight i with hat on top plus straight j with hat on top minus 4 straight k with hat on top right parenthesis. left parenthesis 2 straight i with hat on top plus 6 straight j with hat on top plus 3 straight k with hat on top right parenthesis over denominator vertical line square root of 2 squared plus 6 squared plus 3 squared end root vertical line end fraction end cell row cell equals fraction numerator left square bracket 7 cross times 2 plus 1 cross times 6 plus left parenthesis negative 4 right parenthesis cross times 3 right square bracket over denominator vertical line square root of 2 squared plus 6 squared plus 3 squared end root vertical line end fraction end cell row cell equals fraction numerator left square bracket 14 plus 6 minus 12 right square bracket over denominator vertical line square root of 2 squared plus 6 squared plus 3 squared end root vertical line end fraction end cell row cell equals fraction numerator 8 over denominator vertical line square root of 49 vertical line end fraction equals 8 over 7 end cell end table end style

Solution 37

begin mathsize 12px style table attributes columnalign left end attributes row cell straight a with rightwards arrow on top equals text 2 end text straight i with hat on top plus straight lambda straight j with hat on top plus straight k with hat on top end cell row cell straight b with rightwards arrow on top equals straight i with hat on top minus 2 straight j with hat on top plus 3 straight k with hat on top end cell row cell Since text    end text straight a with rightwards arrow on top text    end text and text    end text straight b with rightwards arrow on top text    end text are text    end text perpendicular text    end text to text    end text each text    end text other end cell row cell straight a with rightwards arrow on top. straight b with rightwards arrow on top equals 0 end cell row cell rightwards double arrow left parenthesis text 2 end text straight i with hat on top plus straight lambda straight j with hat on top plus straight k with hat on top right parenthesis. left parenthesis straight i with hat on top minus 2 straight j with hat on top plus 3 straight k with hat on top right parenthesis equals 0 end cell row cell rightwards double arrow 2 cross times 1 plus straight lambda cross times left parenthesis negative 2 right parenthesis plus 1 cross times 3 equals 0 end cell row cell rightwards double arrow 2 minus 2 straight lambda plus 3 equals 0 end cell row cell rightwards double arrow 5 equals 2 straight lambda end cell row cell therefore straight lambda equals 5 over 2 end cell end table end style

Solution 38

Solution 39

begin mathsize 12px style table attributes columnalign left end attributes row cell Here comma straight a with rightwards arrow on top text    end text and text    end text straight b with rightwards arrow on top text    end text are text    end text perpendicular text    end text to text    end text each text    end text other. end cell row cell straight a with rightwards arrow on top text   end text. straight b with rightwards arrow on top text   end text equals 0 end cell row cell vertical line straight a with rightwards arrow on top plus straight b with rightwards arrow on top vertical line squared equals 13 squared end cell row cell rightwards double arrow left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis equals 169 end cell row cell rightwards double arrow straight a with rightwards arrow on top. straight a with rightwards arrow on top plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight b with rightwards arrow on top. straight a with rightwards arrow on top plus straight b with rightwards arrow on top. straight b with rightwards arrow on top equals 169 end cell row cell rightwards double arrow vertical line straight a with rightwards arrow on top vertical line squared plus 2 straight a with rightwards arrow on top. straight b with rightwards arrow on top plus vertical line straight b with rightwards arrow on top vertical line squared equals 9 end cell row cell rightwards double arrow 5 squared plus 2 cross times 0 plus vertical line straight b with rightwards arrow on top vertical line squared equals 169 end cell row cell rightwards double arrow vertical line straight b with rightwards arrow on top vertical line squared equals 169 minus 25 end cell row cell rightwards double arrow vertical line straight b with rightwards arrow on top vertical line squared equals 144 end cell row cell therefore vertical line straight b with rightwards arrow on top vertical line equals 12 end cell end table end style

Solution 40

begin mathsize 12px style table attributes columnalign left end attributes row cell vertical line straight a with rightwards arrow on top vertical line equals 3 text ,  end text vertical line straight b with rightwards arrow on top vertical line equals 2 over 3 text   end text end cell row cell vertical line straight a with rightwards arrow on top cross times straight b with rightwards arrow on top vertical line equals vertical line vertical line straight a with rightwards arrow on top vertical line vertical line straight b with rightwards arrow on top vertical line sinθ. vertical line straight n with hat on top vertical line vertical line end cell row cell rightwards double arrow 1 equals vertical line 3 cross times 2 over 3 cross times sinθ.1 vertical line end cell row cell rightwards double arrow 1 equals 3 cross times 2 over 3 cross times sinθ end cell row cell rightwards double arrow sinθ equals 1 half end cell row cell therefore straight theta equals straight pi over 6 end cell end table end style

Solution 41

begin mathsize 12px style table attributes columnalign left end attributes row cell vertical line straight a with rightwards arrow on top vertical line equals 1 comma vertical line straight b with rightwards arrow on top vertical line equals 1 comma end cell row cell vertical line straight a with rightwards arrow on top plus straight b with rightwards arrow on top vertical line equals 1 end cell row cell rightwards double arrow vertical line straight a with rightwards arrow on top plus straight b with rightwards arrow on top vertical line squared equals 1 squared end cell row cell rightwards double arrow left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis. left parenthesis straight a with rightwards arrow on top plus straight b with rightwards arrow on top right parenthesis equals 1 end cell row cell rightwards double arrow left parenthesis straight a with rightwards arrow on top. straight a with rightwards arrow on top plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight b with rightwards arrow on top. straight a with rightwards arrow on top plus straight b with rightwards arrow on top. straight b with rightwards arrow on top right parenthesis equals 1 end cell row cell rightwards double arrow vertical line straight a with rightwards arrow on top vertical line squared plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight a with rightwards arrow on top. straight b with rightwards arrow on top plus vertical line straight b with rightwards arrow on top vertical line squared equals 1 end cell row cell rightwards double arrow 1 plus 2 straight a with rightwards arrow on top. straight b with rightwards arrow on top plus 1 equals 1 end cell row cell rightwards double arrow 2 straight a with rightwards arrow on top. straight b with rightwards arrow on top equals negative 1 end cell row cell rightwards double arrow straight a with rightwards arrow on top. straight b with rightwards arrow on top equals negative 1 half end cell row cell rightwards double arrow vertical line straight a with rightwards arrow on top vertical line vertical line straight b with rightwards arrow on top vertical line. cosθ equals negative 1 half end cell row cell rightwards double arrow 1 cross times 1 cross times cosθ equals negative 1 half end cell row cell rightwards double arrow cosθ equals cos fraction numerator 2 straight pi over denominator 3 end fraction end cell row cell therefore straight theta equals fraction numerator 2 straight pi over denominator 3 end fraction end cell end table end style

Solution 42

begin mathsize 12px style table attributes columnalign left end attributes row cell vertical line straight a with rightwards arrow on top vertical line equals 1 comma vertical line straight b with rightwards arrow on top vertical line equals 1 comma end cell row cell vertical line straight a with rightwards arrow on top square root of text 3 end text end root minus straight b with rightwards arrow on top vertical line equals 1 end cell row cell rightwards double arrow vertical line straight a with rightwards arrow on top square root of text 3 end text end root minus straight b with rightwards arrow on top vertical line squared equals 1 squared end cell row cell rightwards double arrow left parenthesis straight a with rightwards arrow on top square root of text 3 end text end root minus straight b with rightwards arrow on top right parenthesis. left parenthesis straight a with rightwards arrow on top square root of text 3 end text end root minus straight b with rightwards arrow on top right parenthesis equals 1 end cell row cell rightwards double arrow 3 straight a with rightwards arrow on top. straight a with rightwards arrow on top minus square root of text 3 end text end root straight a with rightwards arrow on top. straight b with rightwards arrow on top minus square root of text 3 end text end root straight a with rightwards arrow on top. straight b with rightwards arrow on top plus straight b with rightwards arrow on top. straight b with rightwards arrow on top equals 1 end cell row cell rightwards double arrow 3 cross times 1 squared minus 2 square root of text 3 end text end root straight a with rightwards arrow on top. straight b with rightwards arrow on top plus 1 squared equals 1 end cell row cell rightwards double arrow 3 minus 2 square root of text 3 end text end root straight a with rightwards arrow on top. straight b with rightwards arrow on top plus 1 equals 1 end cell row cell rightwards double arrow negative 2 square root of text 3 end text end root straight a with rightwards arrow on top. straight b with rightwards arrow on top equals negative 3 end cell row cell rightwards double arrow straight a with rightwards arrow on top. straight b with rightwards arrow on top equals fraction numerator 3 over denominator 2 square root of 3 end fraction end cell row cell rightwards double arrow vertical line straight a with rightwards arrow on top vertical line. vertical line straight b with rightwards arrow on top vertical line cosθ equals fraction numerator square root of 3 over denominator 2 end fraction end cell row cell rightwards double arrow cosθ equals fraction numerator square root of 3 over denominator 2 end fraction end cell row cell rightwards double arrow cosθ equals cos straight pi over 6 end cell row cell therefore straight theta equals straight pi over 6 end cell end table end style

TopperLearning provides step-by-step solutions for each question in each chapter in the RD Sharma textbook for class 9. Access the CBSE Class 9 Mathematics Chapter 24 - Scalar Or Dot Product for free. The textbook questions have been solved by our subject matter experts to help you understand how to answer them. Our RD Sharma Textbook Solutions will help you to study and revise, and you can easily clear your fundamentals of Chapter 24 - Scalar Or Dot Product.

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CBSE XII Science - Mathematics

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