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Addition of vectors

We are going to determine the resultant of two vectors, 𝐅 ⃗ 1 and 𝐅 ⃗ 2: 1)𝐅1 ⃗ and 𝐅2 ⃗ have the same direction: The magnitude of 𝐅 ⃗ is given by 𝐅  = 𝐅1 + 𝐅2 The direction of F is same as that of F1 and F2. 2) 𝐅1 ⃗ and 𝐅2 ⃗ have opposite directions: The magnitude of 𝐅 ⃗ is given by F = |F1 - F2 | The direction of F is same as that of the larger vector. 3) 𝐅1 ⃗ and 𝐅2 ⃗ are perpendicular to each other: The magnitude of 𝐅 ⃗ is given by Pythagoras' theorem: F = √(〖F1〗^2+ 〖F2〗^2 ) The line of action of 𝐅 ⃗ is along the diagonal of the parallelogram formed by 𝐅 ⃗ 1 and 𝐅 ⃗ 2 The angle between the resultant vector 𝐅 ⃗ and the vector 𝐅2 ⃗ is α, where tan α = Opp/Adj = F1/F2 4) 𝐅1 ⃗ and 𝐅2 ⃗ are concurrent: The magnitude of 𝐅 ⃗ is given by the Law of Cosines (Cosine rule): F^2 = (F1)^2 + (F2)^2 - (2 F1 F2 cosβ) The line of action of 𝐅 ⃗ is along the diagonal of the parallelogram formed by 𝐅1 ⃗ and 𝐅2 ⃗ . 5) To determine the magnitude and the direction of the resultant vector, the component method is preferable. Consider two vectors 𝐅1 ⃗ and 𝐅2 ⃗ . Let 𝐅 ⃗ be their resultant. In this method, we project 𝐅 ⃗1 and 𝐅 ⃗2 onto the x-axis and the y-axis such that: 𝐅 ⃗1 = 𝐅 ⃗ 1x i + 𝐅 ⃗1y j and 𝐅 ⃗ 2 = 𝐅 ⃗2x i + 𝐅 ⃗ 2y j Then: 𝐅 ⃗ = 𝐅 ⃗ 1 + 𝐅 ⃗2 = (F1x + F2x) i + (F1y + F2y) j ; therefore, 𝐅 ⃗ = 𝐅 ⃗ x i + 𝐅 ⃗y j The magnitude of 𝐅 ⃗ is: F = √(〖Fx〗^2+ 〖Fy〗^2 ) The angle between 𝐅 ⃗ and the positive x-axis is α, where tan α = Fy/Fx 6) Scale drawing method (Scale diagram) The resultant vector 𝐅 ⃗ can be determined graphically. Consider the two concurrent vectors, 𝐅 ⃗ 1 and 𝐅 ⃗ 2. To determine their resultant graphically: Choose a convenient drawing scale (1cm → s N) and then represent 𝐅 ⃗ 1 and 𝐅 ⃗ 2. Complete then the parallelogram formed by F1 and F2. Measure the length 𝓁 of the diagonal drawn from point A. The magnitude of F ⃗ is given by: F = 𝓁 × s Use the protractor to measure the angle between 𝐅 ⃗ and 𝐅 ⃗1 or between 𝐅 ⃗ and 𝐅 ⃗ 2.