Project Activity 26.3. Now we look for P...= [b'][b] [b3]]. Assume that the plane p in B+B which the planet's orbit lies has equation z = ax + by.
a) Explain why b = cos(b1) + sin(b2).
(b) The x' axis is the intersection of the plane z = ax + by with the plane z = 0, so the equation of the axis in terms of x and y is x + by = 0. Now we determine the coordinates of b in terms of the basis B.
i. Explain why the vector [b - a 0] lies on the x' axis. We take this vector to point in the positive x' direction. This gives us another representation of b' - namely that
ii. Explain why a vector in the plane z = ax + by orthogonal to b' is [a b a^2 + b^2].
iii. From the previous part we have b^2 =
Let G (0, 0) be the terminal point of the projection of b^2 onto the xy plane. Show that OG is orthogonal to by.
iv. Let θ be the angle between the plane p and the y plane as illustrated at right in Figure 26.3. Explain why ||OG|| = cos(θ). Then explain why
b = [-cos(θ) sin(θ) cos(θ) cos(θ) sin(θ)]^T.
(Hint: Use the trigonometric identities cos(A + B) = -sin(A) and sin(A + B) = cos(A).)