I have an existing 14/3 romex that only has two standard lights on the run. I want to cut this wire, continue that run, but also I want a simple electric outlet in my basement. This is where the 14/2 would come into the picture.
I would tap into the existing 14/3, continue it, but then split off for a new outlet with the 14/2. I want to know if taking only one of the two hots is ok and continuing with two hots to the light switches. Thank you.
Please see picture.
You can splice into a 3-wire cable just as you'd splice a 2-wire cable, assuming it's not for a three-way switch or some other specialty case. Do it as you've drawn.
You'll need slack. If you don't have it, plan for either using two junction boxes, or splice at an existing box.
For the two-box method, mount two boxes at least 24" apart and cut the cable between. Make your splice in whichever box is most convenient and run a jumper between boxes. Obviously this means you need more /3 cable. Do not run parallel cables as a substitute, which would be illegal.
Be aware that junction boxes must remain accessible. Don't bury them in your walls or ceiling, or cover them with cabinets.
There's no such thing as a mid-cable "tap".
Any splices must be done inside a junction box. The box must remain accessible forever. You need at least 6" of wire inside the junction box - 9" is better so you have some margin for screw-ups. Therefore aim for at least 18" of slack.
The only workable way is to have TWO boxes set at least 18" apart along the cable run. (or somewhere else if the length allows, e.g. two boxes 12' apart and down 5 feet). Then, you add additional cable to connect the two boxes to restore the original circuit (e.g. a 24' run that goes up 5 feet, over the 12 feet, and down the other 5 feet).
Again, the boxes must be accessible forever. They cannot be covered up with any building material (a cabinet with hinges and a latched door is fine. Easily lifted drop ceilings are also fine. A lift-out false back to a bookcase is debatable since the electrician will never find the darn thing when trying to chase a wire problem.)
It's perfectly acceptable to bring the splice box somewhere visible and just go ahead and make it a receptacle box. You can have receptacles in weird places. (however if they're GFCI receps that need to be TESTed and RESET sometimes, they must be sanely located).
