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4-2 Components of Vectors Choosing a Coordinate System No correct answer on how to choose Choosing a Correct coordinate system will better allow certain problems to be solved Similar to laying a grid down on a sheet of paper ◦ Choose the center of the grid (origin) ◦ On Earth: generally us the four Directions ◦ Through air: x-axis chosen to be horizontal and y-axis chosen to be vertical Direction of a vector is defined as the angle made with the x-axis measured counterclockwise Components A vector may be broken up into two component vectors. ◦ Parallel with x-axis (Ax) ◦ Parallel with y-axis (Ay) Original Vector A is the sum of the two component vectors ◦ A = Ax + Ay ◦ Vector Resolution is the process of breaking a vector into it’s components Components All algebraic calculations involve only the components of vectors, not the vector itself ◦ Find by using Trigonometry Components are calculated where angle θ is measured counterclockwise from positive x-axis ◦ Ax = A cos θ; cos θ = (adjacent side/hypotenuse) = Ax / A ◦ Ay = A sin θ; sin θ = (opposite side/hypotenuse) = Ay / A Practice A bus travels 23.0 km on a straight road that is 30o N of E. Where are the east and north components of its displacement? Sketch problem Known: A = 23.0 km = 30o Unknown: Ax = ? Ay = ? Calculation ◦ Ax = A cos θ; Ax = (23.0 km)cos 30o ◦ = +19.0 km ◦ Ay = A sin θ; Ay = (23.0 km)sin 30o ◦ = +11.5 km Practice What are the components of a vector of a magnitude 1.5 m at an angle of 35o form the positive x-axis? Algebraic Addition of Vectors Two or more vectors (A, B, C,…) may be added by first resolving each vector to its x & y components ◦ The x-components are added to form the resultant (Rx) ◦ Rx = Ax + Bx + Cx + … ◦ The y-components are added to form the resultant (Ry) ◦ Ry = Ay + By + Cy + … ◦ To find the angle or direction of the resultant, the tangent of the angle that the vector makes with the xaxis is the following (tan θ = Ry/Rx) ◦ Find the angle by using the tan-1 key Algebraic Addition of Vectors Cy C R By Ay A Ax B Bx Cx Rx Ry Practice A GPS reciever told you that your home was 15.0 km at a direction 40o North of West, but the only path led directly north. If you took that path and walked 10.0 km, how far, and in what direction would you have to walk to reach your home? Draw the resultant vector, R from your original location to home Draw A, the known displacement Draw B, the unknown displacement A GPS reciever told you that your home was 15.0 km at a direction 40o North of West, but the only path led directly north. If you took that path and walked 10.0 km, how far, and in what direction would you have to walk to reach your home? Rx = R cos θ ; (15.0 km) cos θ 140o; = -11.5 km Ry = R sin θ ; (15.0 km) cos θ 140o; = +9.6 km Ax = 0.0 km; Ay = 10.0 km R = A + B; so B = R – A Bx = Rx – Ax = (-11.5 km) – (0.0 km) = -11.5 km By = Ry – Ay = (9.6 km) – (10.0 km) = -0.4 km B = √Bx2 + By2 = √(-11.5 km)2 + (-0.4 km)2 = 11.5 km tan θ = By/Bx = (-0.4 km)/(-11.5 km) = +0.035 θ = tan-1(+0.035) = 2.0o B = 11.5 km, 2.0o south of west