Vector ...proof the pythagoras theorem?

the sides of triangle ABC are represented by th Vectors AB=a BC=b CA= c

Prove Pythagoras Theorem IaI^2 + IbI^2 = IcI^2, if angle ABC 90

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  • 1 decade ago
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    We'll use * to denote the dot product ( http://mathworld.wolfram.com/DotProduct.html ).

    By vector addition, c = a + b.

    Now, the square of the magnitude of c is

    | c |^2 = c * c

    = ( a + b ) * ( a + b )

    = a * a + b * b + 2 a * b

    = | a |^2 + | b |^2,

    since a * b = 0 because a is perpendicular to b (the angle between them is 90 degrees). If the angle theta between them is not 90 degrees, you get

    | c |^2 = | a |^2 + | b |^2 + 2 | a | | b | cos(theta),

    the law of cosines.

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  • 4 years ago

    As written above, the notion of "multiverse" is merely a mathematical model based on different attempts to understand the holy grail of science: how to reunite Relativity and Quantum and define the gravitational force. It goes through the way of the String theory, 11 dimensions, etc. The problem, when talking about multiverse or, multiple universes, is that most people sees it as beans in a jar, i.e. co-existing universes. It is not meant as such. When we observe the universe through vision (space) and memory (time) we do it as we would for an object on earth. We should rather use space-time and, as such, there is no center nor edge nor start or end to the universe and if multiverses exist, it should rather be considered as another dimension to time. One can imagine that according Quantum uncertainty, the future isn't yet written and what can happen, does happen ... in another "multiverse." If you remember the metaphor of the ape typing randomly on a typewriter, ending by typing a Shakespeare poem, then we must consider that if time doesn't start and ends, what can happen, will and not only once but an infinite number of times.

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