Pardon me for asking such a dumb question, but how was this length vector calculated? How did we arrive at [0.71, 0.71]?

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PCA Algorithm (optional), 9:10

Andrew has drawn that purple vector at a 45 degree angle to both axes, and its given length is 1 unit. So you can use Pythagoras to compute the length along each axis. 0.71 is a close approximation of the square root of 2.

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