Vectors day 2 Unit vector notation ( i,j,k ) Component Method
Donahue, Steven, Features Editor has reference to this Academic Journal, PHwiki organized this Journal Vectors day 2Unit vector notation (i,j,k)Consider 3D axes (x, y, z)Define unit vectors, i, j, kExamples of Use:40 m, E = 40 i 40 m, W = -40 i30 m, N = 30 j 30 m, S = -30 j20 m, out = 20 k 20 m, in = -20 kExample 4: A woman walks 30 m, W; then 40 m, N. Write her displacement in i,j notation in addition to in R,q notation.R = Rxi + Ry jR = -30 i + 40 jRx = – 30 mRy = + 40 mIn i,j notation, we have:Displacement is 30 m west in addition to 40 m north of the starting position.
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Example 4 (Cont.): Next we find her displacement in R,q notation.q = 126.9o(R,q) = (50 m, 126.9o)R = 50 mq = 1800 59.10Example 6: Town A is 35 km south in addition to 46 km west of Town B. Find length in addition to direction of highway between towns.R = 57.8 kmf = 52.70 S. of W.R = -46 i 35 jExample 7. Find the components of the 240-N as long as ce exerted by the boy on the girl if his arm makes an angle of 280 with the ground.Fx = -(240 N) cos 280 = -212 NFy = +(240 N) sin 280 = +113 N
Example 8. Find the components of a 300-N as long as ce acting along the h in addition to le of a lawn-mower. The angle with the ground is 320.Fx = -(300 N) cos 320 = -254 NFy = -(300 N) sin 320 = -159 NComponent Method2. Draw resultant from origin to tip of last vector, noting the quadrant of the resultant.3. Write each vector in i,j notation.4. Add vectors algebraically to get resultant in i,j notation. Then convert to (R,q).Example 9. A boat moves 2.0 km east then 4.0 km north, then 3.0 km west, in addition to finally 2.0 km south. Find resultant displacement.1. Start at origin. Draw each vector to scale with tip of 1st to tail of 2nd, tip of 2nd to tail 3rd, in addition to so on as long as others.2. Draw resultant from origin to tip of last vector, noting the quadrant of the resultant.Note: The scale is approximate, but it is still clear that the resultant is in the fourth quadrant.
Example 9 (Cont.) Find resultant displacement.3. Write each vector in i,j notation:A = +2 iB = + 4 jC = -3 iD = – 2 j4. Add vectors A,B,C,D algebraically to get resultant in i,j notation. R =-1 i + 2 j1 km, west in addition to 2 km north of origin.Example 9 (Cont.) Find resultant displacement.Now, We Find R, R = 2.24 km = 63.40 N or WReminder of Significant Units:In the previous example, we assume that the distances are 2.00 km, 4.00 km, in addition to 3.00 km.Thus, the answer must be reported as:R = 2.24 km, 63.40 N of W
Significant Digits as long as Anglesq = 36.9o; 323.1oSince a tenth of a degree can often be significant, sometimes a fourth digit is needed.Rule: Write angles to the nearest tenth of a degree. See the two examples below:Example 10: Find R,q as long as the three vector displacements below: 1. First draw vectors A, B, in addition to C to approximate scale in addition to indicate angles. (Rough drawing)2. Draw resultant from origin to tip of last vector; noting the quadrant of the resultant. (R,q)3. Write each vector in i,j notation. (Continued )Example 10: Find R,q as long as the three vector displacements below: (A table may help.)
Example 10 (Cont.): Find i,j as long as three vectors: A = 5 m,00; B = 2.1 m, 200; C = 0.5 m, 900.4. Add vectors to get resultant R in i,j notation.R =Example 10 (Cont.): Find i,j as long as three vectors: A = 5 m,00; B = 2.1 m, 200; C = 0.5 m, 900.R = 7.08 mq = 9.930 N. of E.5. Determine R,q from x,y:Example 11: A bike travels 20 m, E then 40 m at 60o N of W, in addition to finally 30 m at 210o. What is the resultant displacement graphically60o30oRfGraphically, we use ruler in addition to protractor to draw components, then measure the Resultant R,qA = 20 m, EB = 40 mC = 30 mR = (32.6 m, 143.0o)Let 1 cm = 10 m
A Graphical Underst in addition to ing of the Components in addition to of the Resultant is given below:Note: Rx = Ax + Bx + CxBACRy = Ay + By + CyByExample 11 (Cont.) Using the Component Method to solve as long as the Resultant.Write each vector in i,j notation.Ax = 20 m, Ay = 0 Bx = -40 cos 60o = -20 mBy = 40 sin 60o = +34.6 mCx = -30 cos 30o = -26 mCy = -30 sin 60o = -15 mB = -20 i + 34.6 jC = -26 i – 15 jA = 20 iExample 11 (Cont.) The Component MethodAdd algebraically:A = 20 iB = -20 i + 34.6 jC = -26 i – 15 jR = -26 i + 19.6 jq = 143o
Example 11 (Cont.) Find the Resultant.R = -26 i + 19.6 jThe Resultant Displacement of the bike is best given by its polar coordinates R in addition to q.R = 32.6 m; q = 1430Example 12. Find A + B + C as long as Vectors Shown below.Ax = 0; Ay = +5 mBx = +12 m; By = 0Cx = (20 m) cos 350Cy = -(20 m) sin -350R =Example 12 (Continued). Find A + B + C Rx = 28.4 mRy = -6.47 mR = 29.1 mq = 12.80 S. of E.
Vector DifferenceFor vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed be as long as e adding.R = A + BVector DifferenceFor vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed be as long as e adding.Addition in addition to SubtractionR = A + BR = A – B
Example 13. Given A = 2.4 km, N in addition to B = 7.8 km, N: find A B in addition to B A.A – B(2.43 N 7.74 S)5.31 km, SB – A(7.74 N 2.43 S)5.31 km, NRRSummary as long as VectorsA scalar quantity is completely specified by its magnitude only. (40 m, 10 gal)A vector quantity is completely specified by its magnitude in addition to direction. (40 m, 300)Summary Continued:Finding the resultant of two perpendicular vectors is like converting from polar (R, q) to the rectangular (Rx, Ry) coordinates.
Component Method as long as VectorsStart at origin in addition to draw each vector in succession as long as ming a labeled polygon.Draw resultant from origin to tip of last vector, noting the quadrant of resultant.Write each vector in i,j notation (Rx,Ry).Add vectors algebraically to get resultant in i,j notation. Then convert to (R,q).Vector DifferenceFor vectors, signs are indicators of direction. Thus, when a vector is subtracted, the sign (direction) must be changed be as long as e adding.Conclusion of Chapter 3B – Vectors
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