help with circuit design
The below schematic demonstrates a circuit that will do this for you. When the PIR signal is low, the NPN transistor will be off, and no current will flow through the relay. When the PIR signal turns goes high, this causes the transistor to turn on, drawing current through the relay coil turning the relay on. 







simulate this circuit &ndash; Schematic created using CircuitLab

Winny in the comments is correct, you should not use 18Vdc on the 12Vdc rated LEDs. While they may be brighter in the short term, this can lead to a few issues:   

The LEDs will dim much faster over time than if you used 12Vdc.
 There is a chance of overheating, and by extension, fire. 

Now, you also say you have an RGB LED strip, not a simple one color strip. You will need to supply power to all three colors, not just one, if you want a white(ish) color out of them. (This is also assuming this strip is a common cathode model, datasheet/links haven't been provided)

EDIT:
To address @user2913869's concern that the sensor may not be able to drive a single NPN transistor, according to this website, and this datasheet, the actual output of that chip is approximately 10mA @ 5V supply voltage.

EDIT 2:
D6 is what is called a flyback diode. This is a necessary component when using relays (and other inductive loads). Inductive loads don't allow current to change immediately in a circuit. This diode allows current to flow back to the relay, rather than trying to push that current through the transistor, which can break the transistor. To put it  simply, it allows the energy stored in the coil of the relay to dissipate without breaking the transistor.