3 1 Visualize cid 71 we place the tail of vector B cid 71 on the tip of vector A cid 71 and connect the tail of vector A we place the tail of the vector cid 71 B cid 71 on the tip of vector A and then connect the tail cid 71 a To find A B cid 71 Solve cid 71 B with the tip of vector cid 71 cid 71 cid 71 A B A b Since cid 71 of vector A cid 71 B with the tip of vector cid 71 B 3 2 Visualize cid 71 cid 71 a To find A B Solve cid 71 tail with vector sB cid 71 b To find A B tip cid 71 then connect vector cid 71 we place the tail of vector B cid 71 on the tip of vector A and then connect vector cid 71 sA we note that cid 71 cid 71 cid 71 A B A cid 71 B cid 71 We place the tail of vector B cid 71 on the tip of vector A and cid 71 sA tail with the tip of vector cid 71 B 3 3 Visualize points to the left and up so the components xE and yE are negative and positive cid 71 Solve Vector E respectively according to the Tactics Box 3 1 a xE E cos and yE E sin b xE E sin and yE E cos Assess Note that the role of sine and cosine are reversed because we are using a different angle and are complementary angles cid 71 3 4 Visualize The position vector r angle with the in the first quadrant axis x whose magnitude r is 10 m has an x component of 6 m It makes an Solve Using trigonometry r cos or 6 m 10 m cos This gives 53 1 Thus the y component of cid 71 the position vector r is yr xr r sin 10 m sin 53 1 8 m Assess The y component is positive since the position vector is in the first quadrant 3 5 Visualize The figure shows the components vx and vy and the angle Solve We have v sin 40 or 10 m s v sin 40 or v 15 56 m s yv 15 56 m s cos 40 The x component is positive since the position vector is in the fourth quadrant 12 m s cos 40 v xv Thus the x component is Assess 3 6 Visualize We will follow rules in Tactics Box 3 1 Solve respectively cid 71 a Vector r points to the right and down so the components xr and yr are positive and negative cid 71 b Vector v xv cid 71 c Vector a xr r cos 100 m cos 45 70 7 m r sin 100 m sin 45 70 7 m yr points to the right and up so the components xv and yv are both positive v cos 300 m s cos 20 282 m s v sin 300 m s sin 20 103 m s yv has the following components xa a cos 2 5 0 m s cos90 0 m s 2 a sin 2 5 0 m s sin 90 5 0 m s 2 ya Assess The components have same units as the vectors Note the minus signs we have manually inserted according to Tactics Box 3 1 3 7 Visualize We will follow the rules given in Tactics Box 3 1 Solve a 5 cm s sin 90 5 cm s xv yv 5 cm s cos90 0 cm s b xa xF 2 10 m s sin 40 6 4 m s 2 2 10 m s cos 40 7 7 m s 2 ya 50 N cos36 9 50 N sin 36 9 c Assess The components have the same units as the vectors Note the minus signs we have manually inserted according to Tactics Box 3 1 30 N yF 40 N 3 8 Visualize cid 71 The components of the vector C cid 75 Solve For C we have cid 71 D xC and and and the angles are shown 3 15 m cos15 3 04 m and yC 3 15 m sin15 0 815 m cid 75 For D we yD xD 25 6sin 30 25 67cos30 have 12 8 cid 71 Assess The components of the vector C cid 71 are unitless because D following rules of Tactics Box 3 1 cid 71 have the same units as C is without units Note the minus signs we have manually inserted xD and itself yD 22 2 3 9 Visualize Solve The magnitude of the vector is expression for have cid 71 i means that E j and the cid 75 E E 2 2 x E In the E is in quadrant IV The angle is below the positive x axis We 250 V m 125 V m 280 V m y 2 2 tan tan 1 tan 2 63 4 1 E y 1 E x 250 V m 125 V m Assess Since E y E the angle made with the x axis is larger than 45 45 for E x E x y 3 10 Visualize Solve a Using the formulas for the magnitude and direction of a vector we have B 2 4 2 4 5 7 tan 1 1 tan 1 45 4 4 b r 2 cm 2 1 cm 2 2 2 cm tan 1 tan 0 5 26 6 1 c v 10 m s 100 m s 100 5 m s 2 2 tan 1 tan 10 84 3 1 d a 2 2 10 m s 2 2 20 m s 22 4 m s 2 tan 1 tan 0 5 26 6 1 1 2 100 10 10 20 Assess Note that 45 when E E where is the angle made with the x axis On the other hand y x when E E y x 45 3 11 Visualize Solve a Using the formulas for the magnitude and direction of a vector we have A 2 4 6 2 7 21 tan 1 tan 1 5 56 3 1 b r 50 m 2 80 m 2 94 3 m tan 1 tan 58 0 c v 20 m s 40 m s 44 7 m s 2 2 tan 1 tan 2 63 4 1 d a 2 2 2 m s 2 2 6 m s 6 3 m s 2 tan 1 1 tan 0 33 18 4 6 4 r y r x 1 80 m 50 m 40 20 2 6 Assess Note that the angle made with the x axis is smaller than 45 whenever E E x y 45 for E y E and 45 for E x E x y In part d is with the y axis where the opposite of this rule applies 3 12 Visualize cid 71 cid 71 B or cid 71 cid 71 cid 71 cid 71 C A We have C A B which shows how …
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