This could be several things. Any time a vacuum cleaner doesn't start, it can either be caused by a bad cord, the plug on the end of the cord, the switch, or the motor on the vacuum.
Judging by what you said in your repair question that you were using the vacuum and it was working fine, and only when the vacuum came unplugged it stopped working. This definitely sounds like your plug probably went bad. When the vacuum is plugged into the wall, and the cord is pulled tight, this can caused too much tension to be placed on the actual plug that's in the wall. We see this quite frequently, especially from people that pull on the vacuum cord from across the room to unplug it. When the plug has too much stress on it, it will cause the prongs to either bend, or loose their connection on the inside of the plug, preventing the vacuum from getting electricity.
An easy way to test the plug without tearing down the entire vacuum is to just wiggle the prongs at the end of the cord. Are the prongs loose at all? If the prongs are loose, this is a sure sign that your plug has gone bad. If the prongs are tight, then it still may be the plug itself, or possibly the entire cord assembly. Again, because you stated the vacuum was working fine until it became unplugged, I tend to think that the plug is probably bad.
Since the plug is the easiest thing to fix, I would recommend trying to replace the plug first. If this doesn't fix it, it can only be the whole cord assembly (which you'll have a new plug anyways), the switch, or the motor.
This is step by step instructions on how to replace the plug on a vacuum. This was a repair that I helped with on a vacuum that had a three wire cord, but putting a two wire plug on will be the same repair minus the ground wire. Honestly, I use a three wire plug with all of the plugs that I repair, regardless of whether or not the vacuum has two or three wires. Typically the three wire plugs are better quality, and hold the wires better. It won't hurt the vacuum or your outlet if you don't have a wire connected to the third prong. You can pick up an electrical plug at any hardware store, or Home Depot.
Here's how to change the plug:
Step 1. Make sure the vacuum is unplugged. Then cut the existing plug off of your Rainbow, at the very end of the cord, right before the plug molding starts:
Step 2. There are typically three screws that hold the plug part into the base of the plug assembly, remove these screws to separate the two parts of the plug:
Step 3. Slide the cord of your rainbow through the bottom part of the plug so the cord sits about 5-6 inches past the opening of the bottom part of the plug:
Step 4. Carefully cut the outer cord jacket off, about one inch from the end of the cord with a sharp knife. Make sure that you don't cut into the actual wires below, and that you're only cutting the outer part of the cord. If you accidentally cut into the wires, just cut the cord again and start over. This is how you can remove the cord jacket:
Step 5. After removing the cord jacket from the cord, cut the paper filling material off, so you have just the two wires (make sure that there is no copper showing through the wires) are the only thing sticking out. Make sure that you have cut enough cord jacket so the wires have enough room to fit into their respective terminals:
Step 6. Use a pair of wire strippers to strip the insulation off of the wires. Strip the wire insulation about about 1/2 inch down both wires.
Step 7. It's best if you twist the copper wires to keep them from fraying.
Step 8. When using a heavier duty plug like the 3 wire plug shown in this repair, the wire actually uses a clamping device to hold down the wire. Cheaper 2 wire plugs just wrap the wire around a screw, in my opinion the heavier duty three wire cords are much safer, and better:
Step 9. Put the White wire into the clamp that has the silver screw, and the black wire into the clamp with the gold screw, and tighten down the screws that hold the wires into place:
Step 10. Tug firmly on both wires to make sure both are securely fastened inside the terminals. If a wire comes loose, loosen the terminal, put the wire back inside, and then tighten down securely.
Step 11. Usually the 3 screws that secure the top of the plug to the bottom part, are aligned with two screws on the top, and one screw on the bottom. Try and line up the screws with the female receptacles on the bottom part of the plug. There may also be a notch on one (and only one) of the terminals, and similar notch on the bottom part of the plug to help you align the two parts correctly:
Step 12. After aligning the screws on the top part of the plug to the bottom part, tighten each screw until snug, then go back and tighten down completely only after each screw is snug. Screw all three screws until they no longer turn:
Step 13. The last step is tighten the lower part of the plug assembly down on top of the cord. This helps relieve stress on the end of the wire that is connected to the plug prongs, just in case the cord is pulled on, this will prevent the cord from pulling out of the top part of the plug assembly. Make sure that you tighten both screws on the lower portion of the plug as tight as absolutely possible:
And you're done! This is how the completed plug should look when you're finished:
Now just plug the vacuum cleaner into a wall outlet, and see if the vacuum turns on. Hopefully it does, and this fixed your problem. If it didn't, then this means that it can only be the cord itself, the switch (in 10 years I've never replaced a Rainbow E or E2 switch), or the motor. I'm fairly confident from what you explained in your repair question that this will fix your problem. If it doesn't however, please just use the Clarification Request form here on Fixya, and just leave a comment explaining that it didn't fix the problem. I will then follow up on how you can test the motor, the cord, and the switch on your Rainbow.
I hope this actually fixed it however. If this was helpful to you in figuring out what happened to your Rainbow, please be kind and rate my repair as helpful by clicking the thumbs up icons on the repair page. This little gesture really and truly does keep me coming back to Fixya and helping people solve their vacuum problems. I really do appreciate it, and I look forward to helping you further diagnosis your Rainbow should you need it.