Last updated at May 29, 2018 by Teachoo

Transcript

Example 23 Find a unit vector perpendicular to each of the vectors 𝑎 + 𝑏 and 𝑎 − 𝑏 where 𝑎 = 𝑖 + 𝑗 + 𝑘, b = 𝑖 + 2 𝑗 + 3 𝑘 . 𝑎 = 𝑖 + 𝑗 + 𝑘 = 1 𝑖 + 1 𝑗 + 1 𝑘 𝑏 = 𝑖 + 2 𝑗 + 3 𝑘 = 1 𝑖 + 2 𝑗 + 3 𝑘 ( 𝑎 + 𝑏) = (1 + 1) 𝑖 + (1 + 2) 𝑗 + (1 + 3) 𝑘 = 2 𝑖 + 3 𝑗 + 4 𝑘 ( 𝑎 − 𝑏) = (1 − 1) 𝑖 + (1 − 2) 𝑗 + (1 − 3) 𝑘 = 0 𝑖 − 1 𝑗 − 2 𝑘 Now, we need to find a vector perpendicular to both 𝑎 + 𝑏 and 𝑎 − 𝑏, We know that ( 𝑎 × 𝑏) is perpendicular to 𝑎 and 𝑏 Replacing 𝑎 by ( 𝑎 + 𝑏) & 𝑏 by ( 𝑎 − 𝑏) ( 𝒂 + 𝒃) × ( 𝒂 − 𝒃) will be perpendicular to ( 𝒂 + 𝒃) and ( 𝒂 − 𝒃) Let ( 𝑎 + 𝑏) × ( 𝑎 − 𝑏) = 𝑐 𝑐 = 𝑖 𝑗 𝑘2340−1−2 = 𝑖 3×−2−(−1×4) − 𝑗 2×−2−(0×4) + 𝑘 2×−1−(0×3) = 𝑖 −6− −4 − 𝑗 −4−0 + 𝑘 −2−0 = 𝑖 (−6 + 4) − 𝑗 (−4) + 𝑘(−2) = -2 𝑖 + 4 𝑗 − 2 𝑘 ∴ 𝒄 = -2 𝒊 + 4 𝒋 − 2 𝒌 Now, Unit vector of 𝑐 = 1𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑐 × 𝑐 Magnitude of 𝑐 = −22+42+ −22 𝑐 = 4+16+4 = 24 = 2×2×6 = 2 6 Unit vector of 𝑐 = 1 𝑐 × 𝑐 = 12 6 −2 𝑖+4 𝑗−2 𝑘 = 12 6 × 2 − 𝑖+2 𝑗− 𝑘 = −1 6 𝑖 + 2 6 𝑗 − 1 6 𝑘 Therefore, required the unit vector is = −𝟏 𝟔 𝒊 + 𝟐 𝟔 𝒋 − 𝟏 𝟔 𝒌 Note: There are always two perpendicular vectors So, another vector would be = − −1 6 𝑖 + 2 6 𝑗 − 1 6 𝑘 = 𝟏 𝟔 𝒊 − 𝟐 𝟔 𝒋 + 𝟏 𝟔 𝒌 Hence, the perpendicular vectors are −1 6 𝑖 + 2 6 𝑗 − 1 6 𝑘 & 1 6 𝑖 − 2 6 𝑗 + 1 6 𝑘

Chapter 10 Class 12 Vector Algebra

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

Ex 10.3, 16 Important

Example 23 Important You are here

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

About the Author

Davneet Singh

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 8 years. He provides courses for Maths and Science at Teachoo. You can check his NCERT Solutions from Class 6 to 12.