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A The resultant is the sum of two or more vectors. Vectors can be added by moving the tail of one vector to the head of another vector without changing the magnitude or direction of the vector. Vector Addition Note: The red vector R has the same magnitude and direction. Multiplying a vector by a scalar number changes its length but not its direction unless the scalar is negative. V 2V -V If two vectors are added at right angles, the magnitude can be found by using the Pythagorean Theorem R2 = A2 + B 2 and the angle by Opp Tan Adj If two vectors are added at any other angle, the magnitude can be found by the Law of Cosines and the angle by the Law of Sines R 2 A2 B 2 2 AB cos Sin A Sin B Sin C a b c 8 meters 62+82=102 36° 6 meters 10 meters The distance traveled is 14 meters and the displacement is 10 meters at 36º south of east. 6 t a n1 36 8 A hiker walks 3 km due east, then makes a 30° turn north of east walks another 5 km. What is the distance and displacement of the hiker? The distance traveled is 3 km + 5 km = 8 km R 5 km 3 km R2 = 32+52- 2*3*5*Cos 150° R2 = 9+25+26=60 R = 7.7 km s in150 s inθ 7.7 5 The displacement is 7.7 km @ 19° north of east 19 Add the following vectors and determine the resultant. 3.0 m/s, 45 and 5.0 m/s, 135 5.83 m/s, 104 A boat travels at 30 m/s due east across a river that is 120 m wide and the current is 12 m/s south. What is the velocity of the boat relative to shore? How long does it take the boat to cross the river? How far downstream will the boat land? 30 m/s 30 m/s 12 m/s 12 m/s The speed will be 122 302 = 32. 3 m/s @ 21° downstream. The time to cross the river will be t = d/v = 120 m / 30 m/s = 4 s The boat will be d = vt = 12 m/s * 4 s = 48 m downstream. Examples Add the following vectors and determine the resultant. 6.0 m/s, 225 + 2.0 m/s, 90 4.80 m/s, 207.9 Add the following vectors and determine the resultant. 6.0 m/s, 225 + 2.0 m/s, 90 • • • • R2 = 22 + 62 – 2*2*6*cos 45 R2 = 4 + 36 –24 cos 45 R2 = 40 – 16.96 = 23 R = 4.8 m sin 45 sin 4 .8 2 R 17 17 6m 2m R = 4.8 m @ 208 45° Ax A cos Ay A sin A Ax Ay A A A 2 x 2 y and tan 1 Ay Ax Rx v x Ry v y tan 1 Ry Rx R R R 2 x 2 y Vector components is taking a vector and finding the corresponding horizontal and vertical components. Vector resolution A Ax Ay Ax A cos Ay A sin A plane travels 500 km at 60°south of east. Find the east and south components of its displacement. de de= 500 km *cos 60°= 250 km 60° ds 500 km ds= 500 km *sin 60°= 433 km Vector equilibrium Maze Game