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Ex 10.4, 3 - If a unit vector a makes angles pi/3 with i, pi/4

Ex 10.4, 3 - Chapter 10 Class 12 Vector Algebra - Part 2
Ex 10.4, 3 - Chapter 10 Class 12 Vector Algebra - Part 3
Ex 10.4, 3 - Chapter 10 Class 12 Vector Algebra - Part 4
Ex 10.4, 3 - Chapter 10 Class 12 Vector Algebra - Part 5

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Ex 10.4, 3 If a unit vector π‘Ž βƒ— makes angles πœ‹/3 with 𝑖 Μ‚, πœ‹/4 , with 𝑗 Μ‚ & an acute angle ΞΈ with π‘˜ Μ‚ , then find ΞΈ and hence, the components of π‘Ž βƒ— . Let us take a unit vector π‘Ž βƒ— = π‘₯𝑖 Μ‚ + y𝑗 Μ‚ + zπ‘˜ Μ‚ So, magnitude of π‘Ž βƒ— = |π‘Ž βƒ— | = 1 Angle of 𝒂 βƒ— with π’Š Μ‚ = 𝝅/πŸ‘ π‘Ž βƒ— . 𝑖 Μ‚ = |π‘Ž βƒ— ||𝑖 Μ‚ | cos πœ‹/3 (x𝑖 Μ‚ + y𝑗 Μ‚ + zπ‘˜ Μ‚). 𝑖 Μ‚ = 1 Γ— 1 Γ— 1/2 (x𝑖 Μ‚ + y𝑗 Μ‚ + zπ‘˜ Μ‚). (1𝑖 Μ‚ + 0𝑗 Μ‚ + 0π‘˜ Μ‚) = 1/2 (x Γ— 1) + (y Γ— 0) + (z Γ— 0) = 1/2 x + 0 + 0 = 1/2 x = 𝟏/𝟐 Angle of 𝒂 βƒ— with 𝒋 Μ‚ = 𝝅/πŸ’ π‘Ž βƒ— . 𝑗 Μ‚ = |π‘Ž βƒ— ||𝑗 Μ‚ | cos πœ‹/4 (x𝑖 Μ‚ + y𝑗 Μ‚ + zπ‘˜ Μ‚). 𝑗 Μ‚ = 1 Γ— 1 Γ— 1/√2 (x𝑖 Μ‚ + y𝑗 Μ‚ + zπ‘˜ Μ‚). (0𝑖 Μ‚ + 1𝑗 Μ‚ + 0π‘˜ Μ‚) = 1/√2 (x Γ— 0) + (y Γ— 1) + (z Γ— 0) = 1/√2 0 + y + 0 = 1/√2 y = 𝟏/√𝟐 Also, Angle of π‘Ž βƒ— with π‘˜ Μ‚ = ΞΈ π‘Ž βƒ—. π‘˜ Μ‚ = |π‘Ž βƒ— ||π‘˜ Μ‚ |Γ—cos⁑"ΞΈ" (x𝑖 Μ‚ + y𝑗 Μ‚ + zπ‘˜ Μ‚). (0𝑖 Μ‚ + .0𝑗 Μ‚ + 1π‘˜ Μ‚) = 1 Γ— 1 Γ— cos ΞΈ (x Γ— 0) + (y Γ— 0) + (z Γ— 1) = cosΞΈ 0 + 0 + z = cos ΞΈ z = cos ΞΈ Now, Magnitude of π‘Ž βƒ— = √(π‘₯^2+𝑦2+𝑧2) 1 = √((1/2)^2+(1/√2)^2+π‘π‘œπ‘ 2"ΞΈ" ) 1 = √(1/4+1/2+π‘π‘œπ‘ 2"ΞΈ" ) 1 = √(3/4+π‘π‘œπ‘ 2"ΞΈ" ) √(3/4+π‘π‘œπ‘ 2"ΞΈ" ) = 1 (√(3/4+π‘π‘œπ‘ 2"ΞΈ" ))^2 = 12 3/4 + π‘π‘œπ‘ 2" ΞΈ" = 1 π‘π‘œπ‘ 2 "ΞΈ" = 1 βˆ’ 3/4 π‘π‘œπ‘ 2" ΞΈ" = 1/4 cos⁑"ΞΈ" = Β± √(1/4) cos⁑"ΞΈ" = Β± 1/2 Since ΞΈ is given an acute angle So, ΞΈ < 90Β° ∴ ΞΈ is in 1st quadrant And, cos ΞΈ is positive in 1st quadrant= So, cos ΞΈ = 1/2 ∴ ΞΈ = 60Β° = 𝝅/πŸ‘ Also, z = cos ΞΈ = cos 60Β° = 𝟏/𝟐 Hence x = 1/2 , y = 1/√2 & z = 1/2 The required vector π‘Ž βƒ— is 1/2 𝑖 Μ‚ + 1/√2 𝑗 Μ‚ + 1/2 π‘˜ Μ‚ So, components of π‘Ž βƒ— are 𝟏/𝟐 , 𝟏/√𝟐 & 𝟏/𝟐

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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.