The velocity of man is V ⃗ 1 = − V 1 j ⃗ . \vec V_{1}=-V_1\vec j. V 1 = − V 1 j .
Suppose the relative velocity of the wind is
V ⃗ W M 1 = a sin ( α ) i ⃗ + a cos ( α ) j ⃗ \vec V_{WM_1}=a\sin(\alpha)\vec i+a\cos(\alpha)\vec j V W M 1 = a sin ( α ) i + a cos ( α ) j The velocity of the wind is
V ⃗ W = V ⃗ 1 + V ⃗ W M 1 = a sin ( α ) i ⃗ + ( − V 1 + a cos ( α ) ) j ⃗ \vec V_W=\vec V_{1}+\vec V_{WM_1}=a\sin(\alpha)\vec i+(-V_1+a\cos(\alpha))\vec j V W = V 1 + V W M 1 = a sin ( α ) i + ( − V 1 + a cos ( α )) j
The velocity of man is V ⃗ 2 = − V 2 j ⃗ . \vec V_{2}=-V_2\vec j. V 2 = − V 2 j .
Suppose the relative velocity of the wind is
V ⃗ W C 2 = b sin ( β ) i ⃗ + b cos ( β ) j ⃗ \vec V_{WC2}=b\sin(\beta)\vec i+b\cos(\beta)\vec j V W C 2 = b sin ( β ) i + b cos ( β ) j
The velocity of the wind is
V ⃗ W = V ⃗ 2 + V ⃗ W C 2 = b sin ( β ) i ⃗ + ( − V 2 + b cos ( β ) ) j ⃗ \vec V_W=\vec V_{2}+\vec V_{WC_2}=b\sin(\beta)\vec i+(-V_2+b\cos(\beta))\vec j V W = V 2 + V W C 2 = b sin ( β ) i + ( − V 2 + b cos ( β )) j
Then
a sin ( α ) = b sin ( β ) − V 1 + a cos ( α ) = − V 2 + b cos ( β ) \begin{matrix}
a\sin(\alpha) =b\sin(\beta) \\
-V_1+a\cos(\alpha)=-V_2+b\cos(\beta)
\end{matrix} a sin ( α ) = b sin ( β ) − V 1 + a cos ( α ) = − V 2 + b cos ( β )
a = b ⋅ sin ( β ) sin ( α ) V 2 − V 1 = b sin ( β ) ( cot ( β ) − cot ( α ) ) \begin{matrix}
a =b\cdot\dfrac{\sin(\beta)}{\sin(\alpha)} \\
V_2-V_1=b\sin(\beta)(\cot(\beta)-\cot(\alpha))
\end{matrix} a = b ⋅ sin ( α ) sin ( β ) V 2 − V 1 = b sin ( β ) ( cot ( β ) − cot ( α ))
cot ( Θ ) = − V 2 + b cos ( α ) b sin ( β ) \cot(\Theta)=\dfrac{-V_2+b\cos(\alpha)}{b\sin(\beta)} cot ( Θ ) = b sin ( β ) − V 2 + b cos ( α )
b sin ( β ) cot ( Θ ) = − V 2 + b sin ( β ) cot ( β ) b\sin(\beta)\cot(\Theta)=-V_2+b\sin(\beta)\cot(\beta) b sin ( β ) cot ( Θ ) = − V 2 + b sin ( β ) cot ( β )
( V 2 − V 1 ) cot ( Θ ) = − V 2 ( ( cot ( β ) − cot ( α ) ) ) (V_2-V_1)\cot(\Theta)=-V_2((\cot(\beta)-\cot(\alpha))) ( V 2 − V 1 ) cot ( Θ ) = − V 2 (( cot ( β ) − cot ( α )))
+ ( V 2 − V 1 ) cot ( β ) +(V_2-V_1)\cot(\beta) + ( V 2 − V 1 ) cot ( β )
( V 2 − V 1 ) cot ( Θ ) = V 2 cot ( α ) − V 1 cot ( β ) (V_2-V_1)\cot(\Theta)=V_2\cot(\alpha)-V_1\cot(\beta) ( V 2 − V 1 ) cot ( Θ ) = V 2 cot ( α ) − V 1 cot ( β )
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