Unformatted text preview: n to be not divisible by 3.) (b) Use the method of contradiction to prove that √ 3 is not a fraction. Quote your lemma in the proof. Problem 4. Use the 2D Pythagorean Theorem to prove the 3D Pythagorean Theorem. That is, prove that the distance between points (0 , , 0) and ( x,y,z ) equals p x 2 + y 2 + z 2 . (Hint: There are two triangles involved.) Problem 5. The dot product of vectors u = ( u 1 ,u 2 ,...,u n ) and v = ( v 1 ,v 2 ,...,v n ) is deﬁned by u · v := u 1 v 1 + u 2 v 2 + ··· + u n v n . The length k u k of a vector u is deﬁned by k u k 2 := u · u . (a) Prove the formula k uv k 2 = k u k 2 + k v k 22 ( u · v ). (b) Use this formula together with the 2D Pythagorean Theorem and its converse to prove the following statement: “the vectors u and v are perpendicular if and only if u · v = 0.” (Hint: Where is the triangle?)...
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This note was uploaded on 01/08/2012 for the course MATH 461 taught by Professor Armstrong during the Fall '11 term at University of Miami.
 Fall '11
 Armstrong
 Math, Pythagorean Theorem

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