Electrical system died. Had it checked at Auto Zone, and the meter said alternator and voltage regulator are good and voltage regulator failed. Where is the voltage regulator or could it be the computer?

A battery place can test the battery and the alternator to see which one is faulty

otherwise get a volt meter and check the voltages from the alternator without the battery being connected it will be higher at approximately 13.8 volts and from the battery 12 Volts .

What make vehicle an year ? Did you check for battery voltage at the heavier gage wire on the back of the alternator ? The connector at the alternator , how many wires are there ? Should have B+ voltage one or possibly two . Is the charging light (battery symbol ) lit ? You could have a blown fuse or fusible link ! Do you have a DVOM - digital volt ohm meter ? Do you know how to use one ? You need to get a wiring diagram for the charging system for your Vehicle .

unfortunately the alternator is only a part of the charging system and it should be tested by an auto electrician not a parts shop
replace the alternator and then take the vehicle to an accredited auto electrician for a charging system check charge rate should be 14.5-14.8 volts DC at around 1500rpm
they can check the alternator out put , voltage regulator or if the regulator is part of the ECM they will be able to test that as well especially when the system is hot as that is when the problem occurs

I would check charging system, alternator, voltage regulator and good be a bad battery take it to auto zone/ advance they will test your charging system for free

Vehicles: any failing to keep its battery charged.

A vehicle unable to charge its own battery has one of 4 problems:
(a) alternator failure
(b) voltage regulator failure
(c) battery failure
(d) wiring problem between battery and alternator/voltage regulator.

One most modern vehicles (including 2002 Lexus RX300 - 2WD and AWD), the voltage regulator is an integral component of the alternator and is not separately serviceable.

In the US, one can get a free "charging system diagnosis" from the popular auto parts chains: AutoZone, Advance Auto Parts & Pep Boys. You needn't remove any parts from the car to get this diagnostic, since they can attach a diagnostic meter to the charging system in the parking lot. This diagnostic aid will tell you exactly which component has failed - battery, alternator or voltage regulator.

In case the vehicle is immobile, one can DIY (do it yourself) the diagnosis.
(a) inspect the wiring for corrosion/loose connections/loose connectors/etc.
(b) check alternator belt/pulley - if drive belt is properly turning the alternator pulley (no slippage/misrouting/etc.), then the mechanical tests are complete, and you'll need to continue testing the electrical performance of the charging system components.
(c) first component to test: battery
DIY test 1: remove battery from car and put battery on a 120VAC automotive battery charger and charge it fully (or just check it in the car with motor off, since the car's charging system is a type of automotive battery charger).
test parameter: a fully charged lead-acid automotive battery should read 12.45 volts on a VOM/DMM
DIY test 2: disassemble battery out of car after driving it to one of the auto parts chain stores (Advance/AutoZone/Pep Boys) for a free battery test. These testers will test the battery under load, which is not possible with just a DMM.
(d) if wiring is good, and battery tests good under load, then
the failed component is the alternator/voltage regulator - by process of elimination.
(e) DIY test 3: direct alternator/voltage regulator test (car must start and idle successfully to perform this test)
Start the car, and put a VOM/DMM across the terminals of the battery. Since the car is running, you'll be reading the output voltage of the alternator and not the output voltage of the battery. The acceptable ranges for alternator/voltage regulator output are:

ALTERNATOR CHARGING VOLTAGE

Most alternators that are charging properly should produce a voltage of about 13.8 to 14.2 volts at idle with the lights and accessories off. Always refer to the vehicle manufacturer's specifications. Many Asian vehicles, for example, have higher charging voltages of around 15 volts.

When the engine is first started, the charging voltage should rise quickly to about two volts above base battery voltage, then taper off, leveling out at the specified voltage.

The exact charging voltage will vary according to the battery's state of charge, the load on the vehicle's electrical system, and temperature. The lower the temperature the higher the charging voltage, and the higher the temperature the lower the charging voltage. The "normal" charging voltage on a typical application might be 13.9 to 15.1 volts at 77 degrees F. But at 20 degrees F. below zero, the charging voltage might be 14.9 to 15.8 volts. On a hot engine on a hot day, the normal charging voltage might drop to 13.5 to 14.3 volts.

Here are the full specs for installation of the 2002 Lexus RX300 alternator - you may be able to check these specs yourself (with a torque wrench), or pass them along to your mechanic.

Note: the VIN 8th digit should be "F" for the 2002 Lexus RX300 (2WD & AWD)

2002 Lexus RX300 (2WD and AWD) - 3.0L Engine, VIN "F" SFI DOHC

Alternator

Drive belt. Tension the belt to 170-180 lbs. for a new belt or 95-135 lbs. for a used belt.
Adjusting alternator lockbolt. Tighten the bolt to 13 ft.-lbs. (18 Nm).
Alternator pivot bolt. Tighten the bolt to 41 ft.-lbs. (56 Nm).

Glossary of acronyms
--------------------------------
DIY = do it yourself
DMM = Digital Multimeter
DOHC = Dual Overhead Cam
SFI = Sequential Fuel Injection
VOM = Volt Ohmmeter

References
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How to test a Car Alternator - todayifoundout.com

Alternator & Charging System Checks - aa1car.com

If it's not holding a charge you could have a bad battery or the alternator is not putting out enough voltage to keep it charged. Have your battery tested at auto zone or advanced auto. Once the battery is ruled out the next thing to do is check the alternator, this can be done with the alternator installed on the vehicle, if you have a digital multi-meter or volt meter you can check alternator out put at the battery. If your D.C voltage reading is more than 13 volts than your alternator is working. Your transmission issue is probably linked to the electrical issue since it is an electronic transmission, and requires a certain amount of voltage to operate properly.

need to have alternator checked,sounds like the voltage regulator is bad and overcharging system-battery.local advance auto or auto zone can check it for free.

start engine unhook positive cable if it dies its bad alternator or fuesable link between batt and alternator also a volt meter will tell u if its charging a good alternator will charge 13.50 to 14 volts

the voltage regulator is going bad in the altenator

the alternator. have it tested as if it iis failing the car will run off of whatever voltage is left in the battery until it is consumed as the alternator is supposed to recharge the battery and supply the power for the cars components