why Implied Vol (VIX) increase with decrease in Stock Price or vice versa? whereas Vega is positively related with change in option price to change in stock price.
you can find from the CBOE paper above mentioned that the value is pretty much that of a strip of vanillas, weighted by 1/K^2.
Typically if spx spot goes down, then realized vol increases. Together with the increase of realized vol, implied vol gets "re-evaluated" and typically marked higher with a steeper skew etc.
the remark of the implied vol surface will cause the Vix index to go up as per the formula from the CBOE white paper.
So, perhaps the better question would have been: "why does vol spike when markets tank?"
Please read "Volatility's Impact On Market Returns" at http://www.investopedia.com/articles/financial-theory/08/volatility.asp.
It is important to remember that VIX is a volatility index comprised of options and not stocks. It predicts volatility of future prices. It is not a measure of the present stock market.
For a more thorough understanding see the white paper entitle "The CBOE Volatility Index - VIX" at https://www.cboe.com/micro/vix/vixwhite.pdf.