Example 26

Transcript

Example 26 Write all the unit vectors in XY-plane Let the unit vector be 𝑎﷯ We know that 𝑎﷯ = 𝑥 𝑖﷯ + y 𝑗﷯ + z 𝑘﷯ Since the vector is in X-Y plane, there is no Z – coordinate. Hence, 𝑎﷯ = x 𝑖﷯ + y 𝑗﷯ + 0 𝑘﷯ 𝑎﷯ = 𝑥 𝑖﷯ + y 𝑗﷯ Taking a general vector 𝑎﷯, making an angle θ with the x – axis Unit vector in direction of x axis is 𝑖﷯ & in y axis is 𝑗﷯ Given that 𝑎﷯ makes an angle of θ with x-axis So, angle between 𝑎﷯ & 𝑖﷯ is θ Now, we know that, 𝑎﷯ . 𝑏﷯ = 𝑎﷯﷯ 𝑏﷯﷯ cos θ, Putting 𝑎﷯ = 𝑎﷯ , 𝑏﷯ = 𝑖﷯ & θ = θ 𝑎﷯ . 𝑖﷯ = 𝑎﷯﷯ 𝑖﷯﷯ cos θ 𝑎﷯ . 𝑖﷯ = 1 × 1 × cos θ 𝑎﷯ . 𝑖﷯ = cos θ (𝑥 𝑖﷯ + y 𝑗﷯ + 0 𝑘﷯). 𝑖﷯ = cos θ (𝑥 𝑖﷯ + y 𝑗﷯ + 0 𝑘﷯). (1 𝑖﷯ + 0 𝑗﷯ + 0 𝑘﷯) = cos θ 𝑥.1 + y.0 + 0.0 = cos θ x = cos θ Similarly, 𝑎﷯ makes an angle of (90° – θ) with y-axis So, angle between 𝑎﷯ & 𝑗﷯ is (90° – θ) Now, we know that, 𝑎﷯ . 𝑏﷯ = 𝑎﷯﷯ 𝑏﷯﷯ cos θ, Putting 𝑎﷯ = 𝑎﷯ , 𝑏﷯ = 𝑗﷯ & θ = (90° – θ) 𝑎﷯ . 𝑗﷯ = 𝑎﷯﷯ 𝑗﷯﷯ cos (90° – θ) 𝑎﷯ . 𝑗﷯ = 1 × 1 × cos (90° – θ) 𝑎﷯ . 𝑗﷯ = cos (90° – θ) 𝑎﷯ . 𝑗﷯ = sin θ (𝑥 𝑖﷯ + y 𝑗﷯ + 0 𝑘﷯). 𝑗﷯ = sin θ (𝑥 𝑖﷯ + y 𝑗﷯ + 0 𝑘﷯). (0 𝑖﷯ + 1 𝑗﷯ + 0 𝑘﷯) = sin θ 𝑥.0 + y.1 + 0.0 = sin θ y = sin θ So, x = cos 𝜃 and y = sin 𝜃 Thus, 𝑎﷯ = x 𝑖﷯ + y 𝑗﷯ = cos 𝜃 𝑖﷯ + sin 𝜃 𝑗﷯ ∴ 𝒂﷯ = cos 𝜃 𝒊﷯ + sin 𝜃 𝒋﷯ This value will be true in all quadrants Quadrant 1: 0 ≤ θ < 90° Quadrant 2: 90° ≤ θ < 180° Quadrant 3: 180° ≤ θ < 270° Quadrant 4: 270° ≤ θ < 360° So, 0° ≤ θ ≤ 360° i.e. 0 ≤ θ ≤ 2π So, 𝒂﷯ = cos 𝜃 𝒊﷯ + sin𝜃 𝒋﷯ ; for 0 ≤ θ ≤ 2π