g Galileo's telescopes were not of high quality by modern standards. He was able to see the moons of Jupiter, but he never reported seeing features on Mars. Use the small-angle formula to find the angular diameter of Mars when it is closest to Earth. How does that compare with the maximum angular diameter of Jupiter

θ₂/ θ₁= 2.58

Explanation:

In this exercise you are asked to compare the angular diameters of Mars and Jupiter. The angular diameter or angle in radians is

           θ = D / R

where D is the diameter of the body, the distance from Earth to the body of interest and θ is angle in radians

The different distances are tabulated with respect to the Sun

Sun -Earth     1,496 10¹¹ m

Sun- Mars     2.28 10¹¹ m

Sun - Jupiter 7.78 10 m

The Radii of the planets are

Mars     3.37 10⁶ m

Jupiter 6.99 10⁷ m

let's calculate the angles for each body

a) Mars

       θ₁ = 2r / R'

the distance from the ground is

      R ’= D_planet - D_earth

      R ’= 2.28 10¹¹ - 1.496 10¹¹

       R ’= 0.784 10¹¹ m

let's calculate

       θ₁ =

        θ₁ = 8.6 10⁻⁵ radians

b) Jupiter

       R ’= 7.78 10¹¹ - 1.496 10¹¹

      R ’= 6.284 10¹¹ m

let's calculate

       θ₂ =

        θ₂ = 2.22 10⁻⁴ radians

the ratio of the angular diameters is

       θ₂/ θ₁ =

        θ₂/ θ₁= 2.58

no. dot product   =0 when vecs at right angles

i don't understand. explain it clearly