Problem Position Vectors

The position vector from the Sun to Earth is

     r_e = r_e  Ð 0^o (counter-clockwise from vertical)

         = 150 × 10⁶ km  Ð 0^o

The position vector from the Sun to Mars:

     r_m = r_m  Ð α

          = 207 × 10⁶ km  Ð 30^o

The difference between r_e and r_m is the vector from Earth to Mars. Using the head-to-tail method gives,

     r_m = r_e + r_em

This equation can be rearranged to give the unknown position vector r_em,

     r_em = r_m - r_e
         = 207× 10⁶ km  Ð 30^o - 150 × 10⁶ km  Ð 180^o

Subtracting the vectors, gives

     r_em = 107.56 × 10⁶ km  Ð 74.21^o

The problem can also be solved using pure geometry. First, define the position vector from Earth to Mars using the equation

     r_m = r_em  Ð β

Next, form a triangle with the three vectors and use the Law of the Cosine,

     r_em = (r_e² + r_m² - 2r_e r_m cosα )^0.5 

           = 107.56 × 10⁶ km (scalar only, not vector)

Then the internal angle θ, can be determined using the Law of the Sine,

     r_em / sinα = r_em / sinθ

     θ = sin^-1 (r_em sinα /r_em ) = 105.79^o

Finally, the desired angle β is

     β = 180^o - θ = 74.21^o

The position vector from Earth to Mars is

     r_em = 107.56 × 10⁶ km  Ð 74.21^o