Help interpreting a mechanics question Watch

ad absurdum
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Suppose that a particle is projected from sea level on the Earth at latitude \lambda in the Northern hemisphere, at an angle \pi / 4 from the upward vertical and in a Northward direction. Show that when the particle returns to sea level (neglecting the curvature of the Earth's surface), is has been deflected to the East by an amount approximately equal to

\frac{\sqrt{2}\omega |{\bf v}|^3}{3g^2}(3\sin\lambda - \cos\lambda).

I can't really make sense of this, specifically what the "upward vertical" is, and how the particle being projected in a northward direction makes sense if we neglect the curvature of the Earth. I think I know what to do once I have the right components for the vectors.

I've attached what my attempt at a diagram, but I don't think it's right.

Edit - My vectors were g = (0,0,-g), omega = (0, omega cos lambda, omega sin lambda), r_0 = (0, 0, R). Wasn't sure about v, I thought it would just be (0,v,0) but I wasn't getting anywhere with this, and I seemed to be not using this pi/4 anywhere.
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DFranklin
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Upward vertical is going to be perpendicular to the earth's surface - so it's your e3 vector.
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ad absurdum
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That worked. Cheers.
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