I have derived vectors from the coordinates of New York and Lisbon, but now I need to find the vector of a point between the two locations, which has a great circle distance from New York of 2790.101817km (Or 2828.356785km from Lisbon).
New York: (1329, -4638, 4159)
Lisbon: (4908, -792, 3983)
*Vectors rounded to avoid decimals
Any help at all on how to go about getting the XYZ vector for this point would be greatly appreciated.
Let $\mathbf N$ be the vector from the center of the earth to New York.
Let $\mathbf L$ be the vector to Lisbon.
Let $\mathbf X$ be the vector to the unknown point.
Let $\mathbf n$ be the unit vector that is perpendicular to all 3 of them.
Let $d=2790.101817\,\text{km}$ be the distance along the great circle from $\mathbf N$ to $\mathbf X$.
Let $\theta $ be the angle between $\mathbf N$ and $\mathbf X$ in the great circle that connects the 2 of them.
Let $R\approx 6371\,\text{km}$ be the radius of the earth.
Then: \begin{align}\mathbf n &= \frac{\mathbf N \times \mathbf L}{\|\mathbf N \times \mathbf L\|}\\ \theta &= \frac dR \\ \mathbf X &= \mathbf N\cos\theta + (\mathbf n\times\mathbf N)\sin\theta \end{align}