Three vectors , , and , each have a magnitude of 52.0 m and lie in an xy plane. Their directions relative to the positive direction of the x axis are 29.0 ˚, 191 ˚, and 311 ˚, respectively. What are (a) the magnitude and (b) the angle of the vector (relative to the +x direction in the range of (-180°, 180°)), and (c) the magnitude and (d) the angle of in the range of (-180°, 180°)? What are (e) the magnitude and (f) the angle (in the range of (-180°, 180°)) of a fourth vector such that ?

a)   A´= A  

b)   θ₁´ = 29º,  θ₂´ = - 169º ,  θ₃´ = -49º

Explanation:

In this exercise you are asked to give the magnitudes and angles of the vectors from another system of

reference

a) The magnitudes

The magnitude of a vector, the size of which is a scalar, this does not depend on the reference system, since it is obtained by subtracting the coordinates of the end point minus the coordinate of the origin of the vector

          A = - x₀

if the vectors are measured in another reference frame

         x_{f}´ = xx_{f}- U

         x₀´ = x₀ -U

where U is the distance between the two reference frames

         A´ = x_{f}´ - x₀´

we substitute

         A´ = (x_{f} - U) - (x₀-U) = x_{f} - x₀

         A´ = A

 it does not change

b) Angles

The given angles are measured from the positive part of the x axis in a counterclockwise direction, it is asked to give these angles from the x axis

     θ₁ = 29º

     does not change

     θ₁´ = 29º

    θ₂ = 191º

   we measure clockwise

     θ₂´ = θ₂ - 360

     θ₂´ = 191 - 360

     θ₂´ = - 169º

     θ₃ =311º

    we measure clockwise

     θ₃´ = 311 -360

     θ₃´ = -49º

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