Last updated at March 11, 2017 by Teachoo

Transcript

Ex 10.3, 13 (Method 1) If 𝑎 , 𝑏, 𝑐 are unit vectors such that 𝑎 + 𝑏 + 𝑐 = 0, find the value of 𝑎 . 𝑏+ 𝑏 . 𝑐 + 𝑐. 𝑎 . Given 𝑎 , 𝑏, 𝑐 are unit vectors Magnitude of 𝑎 , 𝑏, 𝑐 is 1 So, 𝒂 = 𝒃 = 𝒄 = 1 Also, 𝑎 + 𝑏 + 𝑐 = 0 So, 𝒂 + 𝒃 + 𝒄 = 𝟎 = 0 Now, | 𝒂+ 𝒃+ 𝒄 |2 = ( 𝒂 + 𝒃 + 𝒄) . ( 𝒂 + 𝒃 + 𝒄) = 𝑎. 𝑎 + 𝑎 . 𝑏 + 𝒂 . 𝒄 + 𝒃 . 𝒂 + 𝑏 . 𝑏 + 𝑏 . 𝑐 + 𝑐 . 𝑎 + 𝒄 . 𝒃 + 𝑐 . 𝑐 = 𝑎. 𝑎 + 𝑎 . 𝑏 + 𝒄 . 𝒂 + 𝒂 . 𝒃 + 𝑏 . 𝑏 + 𝑏 . 𝑐 + 𝑐 . 𝑎 + 𝒃 . 𝒄 + 𝑐 . 𝑐 = 𝑎 . 𝑎 + 𝑏 . 𝑏 + 𝑐 . 𝑐 + 2 𝑎. 𝑏 + 2 𝑏. 𝑐 + 2 𝑐. 𝑎 = 𝒂 . 𝒂 + 𝒃 . 𝒃 + 𝒄 . 𝒄 + 2( 𝑎. 𝑏 + 𝑏. 𝑐 + 𝑐. 𝑎) = 𝒂𝟐 + 𝒃𝟐 + 𝒄𝟐 + 2 ( 𝑎. 𝑏 + 𝑏. 𝑐 + 𝑐 . 𝑎) = 12 + 12 + 12 + 2( 𝑎. 𝑏 + 𝑏. 𝑐 + 𝑐. 𝑎) = 1 + 1 + 1 + 2( 𝑎. 𝑏 + 𝑏. 𝑐 + 𝑐. 𝑎) = 3 + 2 ( 𝑎. 𝑏 + 𝑏. 𝑐 + 𝑐. 𝑎) ∴ | 𝑎+ 𝑏+ 𝑐 |2 = 3 + 2 ( 𝑎. 𝑏 + 𝑏. 𝑐 + 𝑐. 𝑎) Now, 𝑎 + 𝑏 + 𝑐 = 0 𝑎 + 𝑏 + 𝑐2 = 0 3 + 2 ( 𝑎. 𝑏 + 𝑏. 𝑐 + 𝑐. 𝑎) = 0 2( 𝑎. 𝑏 + 𝑏. 𝑐 + 𝑐. 𝑎) = −3 ( 𝒂. 𝒃 + 𝒃. 𝒄 + 𝒄. 𝒂) = −𝟑𝟐 Ex 10.3, 13 (Method 2) If 𝑎 , 𝑏, 𝑐 are unit vectors such that 𝑎 + 𝑏 + 𝑐 = 0, find the value of 𝑎 . 𝑏+ 𝑏 . 𝑐 + 𝑐. 𝑎 . Given 𝑎 , 𝑏, 𝑐 are unit vectors So, 𝒂 = 𝒃 = 𝒄 = 1 Also, ( 𝑎 + 𝑏 + 𝑐 ) = 0 Now, 𝒂 . ( 𝒂 + 𝒃 + 𝒄) = 𝑎 . 𝑎 + 𝑎. 𝑏 + 𝑎 . 𝑐 𝑎 . 0 = 𝑎. 𝑎 + 𝑎. 𝑏 + 𝑎. 𝑐 0 = 𝒂. 𝒂 + 𝑎. 𝑏 + 𝑎. 𝑐 0 = 𝒂𝟐 + 𝑎. 𝑏 + 𝑎. 𝑐 0 = 𝑎2 + 𝑎. 𝑏 + 𝑎. 𝑐 0 = 𝑎2 + 𝑎. 𝑏 + 𝒄. 𝒂 0 = 12 + 𝑎. 𝑏 + 𝑐. 𝑎 𝑎. 𝑏 + 𝑐. 𝑎 = −1 Also, 𝒃 . ( 𝒂 + 𝒃 + 𝒄) = 𝑏 . 𝑎 + 𝑏. 𝑏 + 𝑏 . 𝑐 𝑏 . 0 = 𝑏. 𝑎 + 𝑏. 𝑏 + 𝑏. 𝑐 0 = 𝒃. 𝒂 + 𝑏. 𝑏 + 𝑏. 𝑐 0 = 𝒂. 𝒃 + 𝑏. 𝑏 + 𝑏. 𝑐 0 = 𝑎. 𝑏 + 𝒃. 𝒃 + 𝑏. 𝑐 0 = 𝑎. 𝑏 + 𝒃2 + 𝑏 . 𝑐 0 = 𝑎. 𝑏 + 12 + 𝑏 . 𝑐 𝑎. 𝑏 + 𝑏. 𝑐 = −1 Also 𝒄 . ( 𝒂+ 𝒃 + 𝒄) = 𝑐 . 𝑎 + 𝑐 . 𝑏 + 𝑐 . 𝑐 𝑐. 0 = 𝑐. 𝑎 + 𝑐. 𝑏 + 𝑐. 𝑐 0 = 𝑐. 𝑎 + 𝒄. 𝒃 + 𝑐. 𝑐 0 = 𝑐. 𝑎 + 𝒃. 𝒄 + 𝑐. 𝑐 0 = 𝑐. 𝑎 + 𝑏. 𝑐 + 𝒄. 𝒄 0 = 𝑐. 𝑎 + 𝑏. 𝑐 + 𝑐2 0 = 𝑐. 𝑎+ 𝑏 . 𝑐 + 12 𝑐. 𝑎 + 𝑏. 𝑐 = −1 Adding (1), (2) and (3), ( 𝑎. 𝑏 + 𝑐. 𝑎) + ( 𝑎. 𝑏 + 𝑏. 𝑐) + ( 𝑐. 𝑎 + 𝑏. 𝑐) = −1 + (–1) + (–1) 2 𝑎. 𝑏 + 2 𝑐. 𝑎 + 2 𝑏. 𝑐 = −3 2( 𝑎. 𝑏 + 𝑏. 𝑐 + 𝑐. 𝑎) = −3 𝒂. 𝒃 + 𝒃. 𝒄 + 𝒄. 𝒂 = −𝟑𝟐

Ex 10.2, 7
Important

Ex 10.2, 9 Important

Ex 10.2, 10 Important

Ex 10.2, 13 Important

Ex 10.2, 17 Important

Example 14 Important

Example 16 Important

Example 21 Important

Ex 10.3, 2 Important

Ex 10.3, 3 Important

Ex 10.3, 10 Important

Ex 10.3, 13 Important You are here

Ex 10.3, 16 Important

Example 23 Important

Example 24 Important

Example 25 Important

Ex 10.4, 2 Important

Ex 10.4, 5 Important

Ex 10.4, 9 Important

Ex 10.4, 10 Important

Ex 10.4, 11 Important

Example 28 Important

Example 29 Important

Example 30 Important

Misc 6 Important

Misc 12 Important

Misc 13 Important

Misc 15 Important

Misc 19 Important

Class 12

Important Question for exams Class 12

- Chapter 1 Class 12 Relation and Functions
- Chapter 2 Class 12 Inverse Trigonometric Functions
- Chapter 3 Class 12 Matrices
- Chapter 4 Class 12 Determinants
- Chapter 5 Class 12 Continuity and Differentiability
- Chapter 6 Class 12 Application of Derivatives
- Chapter 7 Class 12 Integrals
- Chapter 8 Class 12 Application of Integrals
- Chapter 9 Class 12 Differential Equations
- Chapter 10 Class 12 Vector Algebra
- Chapter 11 Class 12 Three Dimensional Geometry
- Chapter 12 Class 12 Linear Programming
- Chapter 13 Class 12 Probability

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Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 7 years. He provides courses for Mathematics and Science from Class 6 to 12. You can learn personally from here https://www.teachoo.com/premium/maths-and-science-classes/.