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If − v − → and w are linearly independent, show that every vector lying in the plane that contains the two lines through the origin parallel → → → to − v and − w can be expressed as a linear combination of − v and − → − → w . Now show that if u does not lie in this plane, then every → → vector in R3 can be expressed as a linear combination of − u, − v − → and w . → → The two statements above are summarized by saying that − v and − w − → − → − → 2 3 (resp. u , v and w ) span R (resp. R ). Challenge problem: 5.

However, in Book VI, Proposition 31 , Euclid presents a generalization of this theorem: draw any polygon using the hypotenuse as one side; then draw similar polygons using the legs of the triangle; Proposition 31 asserts that the sum of the areas of the two polygons on the legs equals that of the polygon on the hypotenuse. Euclid’s proof of this proposition is essentially the argument given above. 16. The Law of Cosines for an acute angle is essentially given by Proposition 13 in Book II of Euclid’s Elements[27, vol.

29), then we have a triangle △OP Q with angle θ at the origin, and two sides given by → a = |− v|= x21 + y12 + z12 → b = |− w| = x22 + y22 + z22 . The distance formula lets us determine the length of the third side: c = dist(P, Q) = △x2 + △y 2 + △z 2 . 9 Thanks to my student Benjamin Brooks, whose questions helped me formulate the approach here. 4. 29: Determining the Angle θ But we also have the Law of Cosines (Exercise 13): c2 = a2 + b2 − 2ab cos θ or 2ab cos θ = a2 + b2 − c2 . 16) We can compute the right-hand side of this equation by substituting the → → expressions for a, b and c in terms of the entries of − v and − w: a2 + b2 − c2 = (x21 + y12 + z12 ) + (x22 + y22 + z22 ) − (△x2 + △y 2 + △z 2 ).