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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

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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 has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.