Misc 1 - Write down a unit vector in XY-plane, making angle 30

Misc 1 - Chapter 10 Class 12 Vector Algebra - Part 2
Misc 1 - Chapter 10 Class 12 Vector Algebra - Part 3 Misc 1 - Chapter 10 Class 12 Vector Algebra - Part 4

  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise

Transcript

Misc 1 Write down a unit vector in XY-plane, making an angle of 30Β° with the positive direction of x-axis.Let the unit vector be 𝒂 βƒ— π‘Ž βƒ— = π‘₯𝑖 Μ‚ + y𝑗 Μ‚ + zπ‘˜ Μ‚ Since the vector is in XY plane, there is no Z –coordinate. π‘Ž βƒ— = x𝑖 Μ‚ + y𝑗 Μ‚ + 0π’Œ Μ‚ 𝒂 βƒ— = π’™π’Š Μ‚ + y𝒋 Μ‚ Since π‘Ž βƒ— makes an angle 30Β° with the x–axis Also, Unit vector in direction of x axis is 𝑖 Μ‚ & in y axis is 𝑗 Μ‚ Angle with X-axis Since π‘Ž βƒ— makes an angle of 30Β° with x-axis So, angle between 𝒂 βƒ— & π’Š Μ‚ is 30Β° We know that, π‘Ž βƒ— . 𝑏 βƒ— = |π‘Ž βƒ— ||𝑏 βƒ— | cos ΞΈ, Putting π‘Ž βƒ— = π‘Ž βƒ— , 𝑏 βƒ— = 𝑖 Μ‚ & ΞΈ = ΞΈ 30Β° 𝒂 βƒ— .π’Š Μ‚ = |𝒂 βƒ— ||π’Š Μ‚ | cos 30Β° π‘Ž βƒ— .𝑖 Μ‚ = 1 Γ— 1 Γ— cos 30Β° π‘Ž βƒ— . 𝑖 Μ‚ = cos 30Β° (π‘₯𝑖 Μ‚ + y𝑗 Μ‚ + 0π‘˜ Μ‚). 𝑖 Μ‚ = cos 30Β° (π‘₯𝑖 Μ‚ + y𝑗 Μ‚ + 0π‘˜ Μ‚). (1𝑖 Μ‚ + 0𝑗 Μ‚ + 0π‘˜ Μ‚) = cos 30Β° π‘₯.1 + y.0 + 0.0 = cos 30Β° (As π‘Ž βƒ— is unit vector, |π‘Ž βƒ— | = 1 & 𝑖 Μ‚ is a unit vector, |𝑖 Μ‚ | = 1) x = cos 30Β° x = βˆšπŸ‘/𝟐 Angle with Y-axis π‘Ž βƒ— makes an angle of (90Β° – 30Β°) i.e. 60Β° with y-axis So, angle between 𝒂 βƒ— & 𝒋 Μ‚ is 60Β° We know that, π‘Ž βƒ— . 𝑏 βƒ— = |π‘Ž βƒ— ||𝑏 βƒ— | cos ΞΈ, Putting π‘Ž βƒ— = π‘Ž βƒ— , 𝑏 βƒ— = 𝑗 Μ‚ & ΞΈ = 60Β° 𝒂 βƒ— .𝒋 Μ‚ = |𝒂 βƒ— ||𝒋 Μ‚ | cos 60Β° π‘Ž βƒ— .𝑗 Μ‚ = 1 Γ— 1 Γ— cos 60Β° 𝒂 βƒ— .𝒋 Μ‚ = cos 60Β° (π‘₯𝑖 Μ‚ + y𝑗 Μ‚ + 0π‘˜ Μ‚). 𝑗 Μ‚ = cos 60Β° (π‘₯𝑖 Μ‚ + y𝑗 Μ‚ + 0π‘˜ Μ‚). (0𝑖 Μ‚ + 1𝑗 Μ‚ + 0π‘˜ Μ‚) = cos 60Β° π‘₯.0 + y.1 + 0.0 = cos 60Β° y = cos 60Β° y = 𝟏/𝟐 Thus, π‘Ž βƒ— = x𝑖 Μ‚ + y𝑗 Μ‚ 𝒂 βƒ— = βˆšπŸ‘/𝟐 π’Š Μ‚ + 𝟏/𝟐 𝒋 Μ‚

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.