The PS3 requires internet access not digital cable and must be connected with a modem and an ethernet cable or through WiFi connection with cable or DSL wireless signal normally with a wireless router connect to the modem.
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The PlayStation 3 can use either a CAT-5 cable or can connect wirelessly. Typically, it's done wirelessly.
If the hotel has broadband WiFi you don't need the computer just use the PS3 WiFi connection and if it has Ethernet then plug the Ethernet cable in your PS3.
Yes, as long as your VCR has an composite input. Simply plug the composite cables from your PS3 into your VCR. If you plan on playing your PS3 through your VCR this way your VCR must be turned on for it to work.
No, a vga cable is not needed for the xbox 360. However if you would like to hook you xbox 360 to a computer monitor then you would want a vga cable.
Plug in the 3 color coded RCA style plugs or purchase a HDMI cable. It's not the brand of TV that matters, but the connections that are in the back of the set. If you don't have any spots to plug anything into buy a new TV to go with your PS3