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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


Transcript

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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.