Three vectors a , D , and C , each have a magnitude of 41.0 m and lie in an xy p

Question

Three vectors a , D , and C , each have a magnitude of 41.0 m and lie in an xy plane. Their directions relative to the positive direction of the x axis are 34.0 °, 190 °, and 314 °, respectively. What are (a) the magnitude and (b) the angle of the vectorb+ C(relative to the +x direction in the range of (-180°, 180)), and () the magnitude and (d) the angle of b+in the range of (-1809, 180°) What are (e) the magnitude and () the angle (in the range of -180, 180°) of a fourth vectord (a b)-(cd) 0? such that (a) Number (b) Number (c) Number (d) Number e) Number f) Number Unit Unit Unit Unit Unit Unit

Explanation / Answer

Given magnitudes of the vectors are of 41 m

making angles with x axis are a 34 degrees b 190 degrees c 314 degrees

so we can write the vectors as  

a = ax i + ay j

ax = a cos theta and ay = a sin theta

a = 41 cos34 I + 41 sin34 = 34 m i + 23 m j

so as  

b = 41 cos190 I + 41 sin190 and = -40.4 m i -7.12 m j

c = 41 cos314 I + 41 sin314 = 28.48 m i -29.5 m j

now the vector sum

a) a + b + c = (34 m i + 23 m j )+(-40.4 m i -7.12 m j) + (28.48 m i -29.5 m j)

= (34-40.4+28.48) m i +(23-7.12-29.5) m j

= (22.08) m i - (13.62) m j

the magnitudei s  

= sqrt((22.08)^2 +(-13.62)^2) m = 25.94 m

the direction is theta = arc tan(-13.62/22.08) = -31.67degrees

c) a - b + c

= (34 m i + 23 m j )-(-40.4 m i -7.12 m j) + (28.48 m i -29.5 m j)

= (34+40.4+28.48) m i +(23+7.12-29.5) m j

= (102.88) m i + (0.62) m j

the magnitudei s  

= sqrt((102.88)^2 +(0.62)^2) m = 102.88 m

the direction is theta = arc tan(0.62/102.88) = 0.34528 degrees

e ) let vector d = dx i + dy j

so (a+b) - (c+d) = 0

(a+b) = (c+d)

(34 m i + 23 m j )+(-40.4 m i -7.12 m j)= (28.48 m i -29.5 m j) +(dx i +dy j )

(34-40.4-28.48)i +(23-7.12+29.5) j = dx i + dy j

-34.88 m i +45.38 m j = dx i +dy j

d = -34.88 m i +45.38 m j  

the magnitude is  

d = sqrt((-34.88)^2+(45.38)^2) m = 57.24 m

the dierection is teta= arc tan (45.38/(-34.88)) = -52.45 degrees

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