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I assume that air resistance force is parallel to the velocity vector but in opposite direction. We have: $$a_x=10\cos\theta$$ $$a_y=10\sin\theta-g\;\Longrightarrow\;a_y+g=10\sin\theta$$ $$\tan\theta=\large{\frac{v_y}{v_x}}$$ Then, $$\large{\frac{v_y}{v_x}}=\large{\frac{a_y+g}{a_x}}\;\Longrightarrow\;\large{\frac{a_y+g}{v_y}}=\large{\frac{a_x}{v_x}}=C\;\... 2 Comments to the post (v2): Ref. 1 is considering the d-dimensional real Euclidean space (\mathbb{R}^d,|\cdot|^2) with the standard norm$$|x|^2~:=~\sum_{\mu=1}^d (x^{\mu})^2~=~\sum_{\mu,\nu=1}^d x^{\mu}\eta_{\mu\nu}x^{\nu}, \qquad \eta_{\mu\nu} ~=~{\rm diag}(1,\ldots, 1),\tag{A}$$and inner product$$\langle x ,y\rangle~:=~\sum_{\mu,\nu=1}^d x^{\mu}\...