NASA is planning an unmanned mission to Mars. Mission planners need the precise position of the planet relative to Earth on the launch date in order to determine the proper velocity for the spacecraft. If the distance or angle is incorrect, the spacecraft could be lost.

What is known:

The radius of Earth's orbit r_{e} is 150 x 10^{6}
km.

The radius of Mars's orbit r_{m} is 207 x 10^{6} km.

The angle α between Earth and Mars relative
to the Sun is 30°.

Assume both Earth's and Mars' orbits around the Sun is circular.

Question

Problem Diagram

What is the position of Mars relative to Earth on the launch day?

Approach

Draw a diagram and label the key points and known position vectors.

Determine the relationship between the unknown position vector and those that are given.

Using geometry, solve for the length and angle of the unknown vector.