Manipulating two graphics sets with the same variable

Question

I have two graphics for a visualization that both depend on the same parameter (an angle). I'd like for both graphics to be controlled by the same slider, but I can't have the graphics overlay each other using Show as they need to stay separate. Any idea how to do this?

 "p2=VectorPlot[{(3Cos[ArcTan[x/y]]Sin[ArcTan[x/y]]),(2Cos[ArcTan[x/y]]\
 Cos[ArcTan[x/y]]-Sin[ArcTan[x/y]]Sin[ArcTan[x/y]])},{x,-2,2},{y,-2,2}]"
 Manipulate[Show[
 p2, Graphics[{Thick, Arrow[{{1 - 0.5 Sin[\[Theta]],  0.5 - 0.5 Cos[\[Theta]] - 0.5}, {1 + 0.5 Sin[\[Theta]], 0.5 + 0.5 Cos[\[Theta]] - 0.5}}],Arrow[{{0, -0.5}, {0, 0.5}}]}, Axes -> False]], {\[Theta], 0, 
 2 Pi}]
 "p1=Plot[Cos[\[Theta]],{\[Theta],0,2 Pi}]"
 Manipulate[
 Show[p1, Graphics[{PointSize[Large], Red, Point[{\[Theta], Cos[\[Theta]]}]}]], {\[Theta], 0, 2 Pi}]

Answer 1

One way is to place both the Shows inside a Column within Manipulate (not quite clear to me if the two graphics in a column is acceptable to you).

p1 = Plot[Cos[\[Theta]], {\[Theta], 0, 2 Pi}];
p2 = VectorPlot[{
   (3 Cos[ArcTan[x/y]] Sin[ArcTan[x/y]]),
   (2 Cos[ArcTan[x/y]] Cos[ArcTan[x/y]] - 
     Sin[ArcTan[x/y]] Sin[ArcTan[x/y]])
   },
  {x, -2, 2},
  {y, -2, 2}
  ];

and then the Manipulate

Manipulate[
 Column[
  {
   Show[
    p2,
    Graphics[
     {
      Thick,
      Arrow[{
        {1 - 0.5 Sin[\[Theta]], 0.5 - 0.5 Cos[\[Theta]] - 0.5},
        {1 + 0.5 Sin[\[Theta]], 0.5 + 0.5 Cos[\[Theta]] - 0.5}
        }
       ],
      Arrow[
       {
        {0, -0.5}, {0, 0.5}
        }
       ]
      },
     Axes -> False
     ],
    ImageSize -> 400
    ],
   Spacer[10],
   Show[
    p1,
    Graphics[
     {
      PointSize[Large],
      Red,
      Point[{\[Theta], Cos[\[Theta]]}]
      }
     ],
    ImageSize -> 400
    ]
   }
  ],
 {\[Theta], 0, 2 Pi}
 ]

If you truly want them separated one can take control by using Dyanmic.

{Slider[Dynamic[\[Theta]], {0, 2 \[Pi]}], Dynamic[\[Theta]]}

and then

Dynamic[
 Show[
  p2,
  Graphics[
   {
    Thick,
    Arrow[{
      {1 - 0.5 Sin[\[Theta]], 0.5 - 0.5 Cos[\[Theta]] - 0.5},
      {1 + 0.5 Sin[\[Theta]], 0.5 + 0.5 Cos[\[Theta]] - 0.5}
      }
     ],
    Arrow[
     {
      {0, -0.5}, {0, 0.5}
      }
     ]
    },
   Axes -> False
   ],
  ImageSize -> 400
  ]
 ]

Dynamic[
 Show[
  p1,
  Graphics[
   {
    PointSize[Large],
    Red,
    Point[{\[Theta], Cos[\[Theta]]}]
    }
   ],
  ImageSize -> 400
  ]
 ]

Note: The newlines and spacing I use in the code are completely unnecessary. I just do that to help myself keep track of the level.