Homework Statement

Let vectors from the center of the earth to two points on the surface be ##\vec{r}_1,\vec r_2##. If the angle between them is ##\theta##, find the expression for ##\theta## in terms of the co-ordinates of the two points, them being their latitudes ##\lambda_1,\lambda_2## and ##\phi_1,\phi_2##. (See sketch below)

Relevant Equations

1. A vector ##\vec a = a(\cos\alpha\hat i+\cos\beta\hat j+\cos\gamma\hat k)##, where the angles are well known. Here, ##\alpha = \phi## and ##\gamma = \pi/2-\lambda##.
2. If the direction cosines written in (1) above be given by the symbols ##l,m,n##, then ##\cos\theta=l_a l_b+m_a m_b+n_a n_b##.
3. We also have ##l^2+m^2+n^2=1## as the relation between the direction cosines for a vector.

Attempt : I took the liberty to draw a better image which show the latitudes and longitudes of the two places.

The image shows places ##P_1## and ##P_2## drawn in purple and green and green, respectively, as well as their latitudes ##\lambda_i## and longitudes ##\phi_i## for ##i = 1,2##. The angle between their position vectors is ##\theta##, shown in red.
Cluttered though the symbols are, I hope they can be distinguished.

We know ##\cos\theta = l_1 l_2+m_1 m_2 + n_1 n_2\qquad (1).##

Here, ##l_i = \cos\phi_i##, ##n_i = \cos\gamma_i = \sin\lambda_i##, where ##\gamma## is the (well-known) angle that a vector makes with the ##z-##axis. The values ##m_i=\cos\beta_i##, the angle with the ##y-##axis. These can be calculated using ##l^2+m^2+n^2=1\Rightarrow m_i = \sqrt{1-\cos^2\phi_i-\sin^2\lambda_i}##, the index ##i## taking values 1 and 2 throughout.

We know, ##1-\cos^2\phi_i = \sin^2\phi_i##.

From ##(1)## above, all this yields ##\boxed{\cos\theta = \cos\phi_1 \cos\phi_2+\sqrt{\sin^2\phi_1-\sin^2\lambda_1}\sqrt{\sin^2\phi_2-\sin^2\lambda_2}+\sin\lambda_1 \sin\lambda_2}##.

I'd check the answer presently, but am I on the right track?

Many thanks.

I would rotate the sphere so that one point is at the north pole (0,0,1). Then ##\theta## is just the colatitude of the image of the second point.

I would write two unit vectors $$\mathbf{\hat r}_i=\cos\phi_i\cos\lambda_i~\mathbf{\hat x}+\sin\phi_i\cos\lambda_i~\mathbf{\hat y}+\sin\lambda_i~\mathbf{\hat z}~~~~(i=1,2)$$ then take their dot product.

Sorry for coming in late. I would use the method you proposed @kuruman presently, but please tell me if there's something wrong with my solution above. (Post #1)

I think it would be better if I put the problem statement again.

Of course it's crucial what I am assuming as "co-ordinates of the points". The ones given are ##\vec r_1,\vec r_2##, but these are vectors. In their explanation of the problem, see image to the right, they talk of the latitude ##\lambda## and longtidue ##\phi## of a place on the surface of the earth.
I assumed, by this, the coordinates of the two places are ##(\lambda_i,\phi_i)## for ##i=1,2## and obtained this answer for the angle ##\theta## between the two places :
##\small{\boxed{\cos\theta = \cos\phi_1 \cos\phi_2+\sqrt{\sin^2\phi_1-\sin^2\lambda_1}\sqrt{\sin^2\phi_2-\sin^2\lambda_2}+\sin\lambda_1 \sin\lambda_2}}##.

I'd like to know where's the mistake.