When I turn on white lights, these chairs cast a shadow. However, when blue LED lights are turned on the shadow turns yellow. Why is this? Is this due to interference of light or is it just an illusion?
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It is likely to be an illusion -- on the colour wheel, blue is opposite to yellow.
Your eyes adjust to the presence of the blue light (recalibrating, in a way), and the absence of blue is seen as yellow. This is probably also true for the white-balance algorithm on your camera.
Our eyes have two different light receptors, called cones and rods. The cones come in three variants that are sensitive to light in the red, green, and blue wavelength range, respectively, of the color spectrum. Thus, you could say that it is an illusion, in the sense that all colors that are not either red, green, or blue are our brains' way of representing light outside these wavelength ranges, or at intermediate ranges.
If all the cones are triggered, your brain will tell you that the light is "white". If only the blue and red, but not green, are triggered, you will perceive this as "magenta", while if only the red and green, but not blue, receptors are triggered, your brain will tell you that you are seeing "yellow". It makes sense if you take a look at this figure:
When you turn on your blue LEDs, you add blue to the white. In the shadow, however, you only see the light of your white light, but firstly, your white light may be quite yellowish (if it's a "normal" incandescent bulb), and secondly, when the shadow is surrounded by blue light, your brain tries to make it more discernable by "overcompensating" its color to make it the constrast color of blue, i.e. yellow. It's the same effect that makes the rings in the figure below appear colored, even though they're both gray (you can convince yourself of this by zooming in on the rings):
If instead of the white light you have a red, a green, and a blue lamp shining on a white screen from slightly different direction, you can have really cool colored shadows.