3D Vector

Direction

{POx + Cos[Yh] Cos[Zh], POy + Cos[Yh] Sin[Zh], POz - Sin[Yh], 1}

{POx + Cos[Zh] Sin[Xh] Sin[Yh] - Cos[Xh] Sin[Zh], POy + Cos[Xh] Cos[Zh] + Sin[Xh] Sin[Yh] Sin[Zh], POz + Cos[Yh] Sin[Xh], 1}

{POx + Cos[Xh] Cos[Zh] Sin[Yh] + Sin[Xh] Sin[Zh], POy - Cos[Zh] Sin[Xh] + Cos[Xh] Sin[Yh] Sin[Zh], POz + Cos[Xh] Cos[Yh], 1}

Multiple

{POx + DX Cos[Yh] Cos[Zh], POy + DX Cos[Yh] Sin[Zh], POz - DX Sin[Yh], 1}

{POx + DY Cos[Zh] Sin[Xh] Sin[Yh] - DY Cos[Xh] Sin[Zh], POy + DY Cos[Xh] Cos[Zh] + DY Sin[Xh] Sin[Yh] Sin[Zh], POz + DY Cos[Yh] Sin[Xh], 1}

{POx + DZ Cos[Xh] Cos[Zh] Sin[Yh] + DZ Sin[Xh] Sin[Zh], POy - DZ Cos[Zh] Sin[Xh] + DZ Cos[Xh] Sin[Yh] Sin[Zh], POz + DZ Cos[Xh] Cos[Yh], 1}

{POx + DX Cos[Yh] Cos[Zh], POy + DX Cos[Yh] Sin[Zh], POz - DX Sin[Yh], 1}

{POx + DY Cos[Zh] Sin[Xh] Sin[Yh] - DY Cos[Xh] Sin[Zh], POy + DY Cos[Xh] Cos[Zh] + DY Sin[Xh] Sin[Yh] Sin[Zh], POz + DY Cos[Yh] Sin[Xh], 1}

{POx + DZ Cos[Xh] Cos[Zh] Sin[Yh] + DZ Sin[Xh] Sin[Zh], POy - DZ Cos[Zh] Sin[Xh] + DZ Cos[Xh] Sin[Yh] Sin[Zh], POz + DZ Cos[Xh] Cos[Yh], 1}

dX = (PXx^2 + PXy^2)^(1/2)

Zh = ArcTan[PXx, PXy]

Tan[Zh] = PXy/PXx = PXy/dX/PXx/dX = Sin[Zh]/Cos[Zh]

dX = (PXx^2 + PXy^2)^(1/2)

dX^2 = PXx^2 + PXy^2

dX = (PXx^2 + PXy^2)/dX = PXx^2/dX + PXy^2/dX = PXx/dXPXx + PXy/dXPXy

= Cos[Zh] PXx + Sin[Zh] PXy

DX = (PXx^2 + PXy^2 + PXz^2)^(1/2)

Yh = ArcTan[Cos[Zh] PXx + Sin[Zh] PXy, -PXz]

Tan[Yh] = -PXz/dX = -PXz/DX/dX/DX = Sin[Yh]/Cos[Yh]

DX = (dX^2 + (-PXz)^2)^(1/2)

DX^2 = dX^2 + (-PXz)^2

DX = (dX^2 + (-PXz)^2)/DX = dX^2/DX + (-PXz)^2/DX = dX/DXdX + (-PXz)/DX (-PXz)

= Cos[Yh] dX + Sin[Yh] (-PXz)

= Cos[Yh] {(PXx^2 + PXy^2)/dX} + Sin[Yh] (-PXz)

= Cos[Yh] {PXx/dXPXx + PXy/dXPXy} + Sin[Yh] (-PXz)

= Cos[Yh] {Cos[Zh] PXx + Sin[Zh] PXy} + Sin[Yh] (-PXz)

= Cos[Yh] Cos[Zh] PXx + Cos[Yh] Sin[Zh] PXy + Sin[Yh] (-PXz)

= Cos[Yh] Cos[Zh] PXx + Cos[Yh] Sin[Zh] PXy - Sin[Yh] PXz

Distance

Zh = ArcTan[x, y]

Yh = ArcTan[Cos[Zh] x + Sin[Zh] y, z]

D = (d^2 + z^2)^(1/2)

D^2 = d^2 + z^2

D = (d^2 + z^2)/D = d^2/D + z^2/D = d/Dd + z/Dz

= Cos[Yh] d + Sin[Yh] z

= Cos[Yh] {(x^2 + y^2)/d} + Sin[Yh] z

= Cos[Yh] {x/dx + y/dy} + Sin[Yh] z

= Cos[Yh] {Cos[Zh] x + Sin[Zh] y} + Sin[Yh] z

= Cos[Yh] Cos[Zh] x + Cos[Yh] Sin[Zh] y + Sin[Yh] z

∴

{POx + D Cos[Yh] Cos[Zh], POy + D Cos[Yh] Sin[Zh], POz - D Sin[Yh]}

Zh = ArcTan[x, y]

Yh = ArcTan[Cos[Zh] x + Sin[Zh] y, z]

D = Cos[Yh] Cos[Zh] x + Cos[Yh] Sin[Zh] y + Sin[Yh] z

Ex. Formula and solve

1

{d Cos[θ] Cos[φ], d Cos[φ] Sin[θ], d Sin[φ]} = {x, y, z}

d = (x - x0)/ Cos[θ] Cos[φ]

d = (y - y0)/Cos[φ] Sin[θ]

d = (z - z0)/Sin[φ]

S = Simplify[Solve[{(x - x0)/ Cos[θ] Cos[φ] == (y - y0)/Cos[φ] Sin[θ] == (z - z0)/Sin[φ], z == h}, {x, y, z}][[1]]] ;

{x, y, z}/.S

2

d = (x - x0)/(-Cos[t] Sin[θ] Tan[φ] + Cos[θ] (Cos[φ] - Sin[t] Sin[φ] Tan[φ]))

d = (y - y0)/(Cos[φ] Sin[θ] + (Cos[t] Cos[θ] - Sin[t] Sin[θ] Sin[φ]) Tan[φ])

d = (z - z0)/(Sin[φ] + Cos[φ] Sin[t] Tan[φ])

{x//InputForm, y//InputForm, z//InputForm}/.S[[1]]

3

Simplify[Transpose[({{Cos[Th], -Sin[Th], 0}, {Sin[Th], Cos[Th], 0}, {0, 0, 1}}) . ({{Cos[Jh], 0, -Sin[Jh]}, {0, 1, 0}, {Sin[Jh], 0, Cos[Jh]}}) . ({{D}, {0}, {D Tan[Ph]}})][[1]]] + {x0, y0, z0}

4

{Cos[ThH] Cos[JhH], Cos[JhH] Sin[ThH], Sin[JhH]}

d = (x - x0)/(-Cos[t] Sin[θ] Tan[φ] + Cos[θ] (Cos[φ] - Sin[t] Sin[φ] Tan[φ]))

d = (y - y0)/(Cos[φ] Sin[θ] + (Cos[t] Cos[θ] - Sin[t] Sin[θ] Sin[φ]) Tan[φ])

d = (z - z0)/(Sin[φ] + Cos[φ] Sin[t] Tan[φ])

{x//InputForm, y//InputForm, z//InputForm}/.S[[1]]