Example 26 - Write all unit vectors in XY-plane - Class 12 Vector - Unit vector

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  1. Chapter 10 Class 12 Vector Algebra
  2. Serial order wise
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Example 26 Write all the unit vectors in XY-plane Let the unit vector be π‘Žο·― We know that π‘Žο·― = π‘₯ 𝑖﷯ + y 𝑗﷯ + z π‘˜ο·― Since the vector is in X-Y plane, there is no Z – coordinate. Hence, π‘Žο·― = x 𝑖﷯ + y 𝑗﷯ + 0 π‘˜ο·― π‘Žο·― = π‘₯ 𝑖﷯ + y 𝑗﷯ Taking a general vector π‘Žο·―, making an angle ΞΈ with the x – axis Unit vector in direction of x axis is 𝑖﷯ & in y axis is 𝑗﷯ Given that π‘Žο·― makes an angle of ΞΈ with x-axis So, angle between π‘Žο·― & 𝑖﷯ is ΞΈ Now, we know that, π‘Žο·― . 𝑏﷯ = π‘Žο·―ο·― 𝑏﷯﷯ cos ΞΈ, Putting π‘Žο·― = π‘Žο·― , 𝑏﷯ = 𝑖﷯ & ΞΈ = ΞΈ π‘Žο·― . 𝑖﷯ = π‘Žο·―ο·― 𝑖﷯﷯ cos ΞΈ π‘Žο·― . 𝑖﷯ = 1 Γ— 1 Γ— cos ΞΈ π‘Žο·― . 𝑖﷯ = cos ΞΈ (π‘₯ 𝑖﷯ + y 𝑗﷯ + 0 π‘˜ο·―). 𝑖﷯ = cos ΞΈ (π‘₯ 𝑖﷯ + y 𝑗﷯ + 0 π‘˜ο·―). (1 𝑖﷯ + 0 𝑗﷯ + 0 π‘˜ο·―) = cos ΞΈ π‘₯.1 + y.0 + 0.0 = cos ΞΈ x = cos ΞΈ Similarly, π‘Žο·― makes an angle of (90Β° – ΞΈ) with y-axis So, angle between π‘Žο·― & 𝑗﷯ is (90Β° – ΞΈ) Now, we know that, π‘Žο·― . 𝑏﷯ = π‘Žο·―ο·― 𝑏﷯﷯ cos ΞΈ, Putting π‘Žο·― = π‘Žο·― , 𝑏﷯ = 𝑗﷯ & ΞΈ = (90Β° – ΞΈ) π‘Žο·― . 𝑗﷯ = π‘Žο·―ο·― 𝑗﷯﷯ cos (90Β° – ΞΈ) π‘Žο·― . 𝑗﷯ = 1 Γ— 1 Γ— cos (90Β° – ΞΈ) π‘Žο·― . 𝑗﷯ = cos (90Β° – ΞΈ) π‘Žο·― . 𝑗﷯ = sin ΞΈ (π‘₯ 𝑖﷯ + y 𝑗﷯ + 0 π‘˜ο·―). 𝑗﷯ = sin ΞΈ (π‘₯ 𝑖﷯ + y 𝑗﷯ + 0 π‘˜ο·―). (0 𝑖﷯ + 1 𝑗﷯ + 0 π‘˜ο·―) = sin ΞΈ π‘₯.0 + y.1 + 0.0 = sin ΞΈ y = sin ΞΈ So, x = cos πœƒ and y = sin πœƒ Thus, π‘Žο·― = x 𝑖﷯ + y 𝑗﷯ = cos πœƒ 𝑖﷯ + sin πœƒ 𝑗﷯ ∴ 𝒂﷯ = cos πœƒ π’Šο·― + sin πœƒ 𝒋﷯ This value will be true in all quadrants Quadrant 1: 0 ≀ ΞΈ < 90Β° Quadrant 2: 90Β° ≀ ΞΈ < 180Β° Quadrant 3: 180Β° ≀ ΞΈ < 270Β° Quadrant 4: 270Β° ≀ ΞΈ < 360Β° So, 0Β° ≀ ΞΈ ≀ 360Β° i.e. 0 ≀ ΞΈ ≀ 2Ο€ So, 𝒂﷯ = cos πœƒ π’Šο·― + sinπœƒ 𝒋﷯ ; for 0 ≀ ΞΈ ≀ 2Ο€

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