By Frenkel D., Portugal R.

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V + u . w Distributivity ( cu) . v = c ( u . v) u · u 2: 0 and u · u = 0 if and only if u = 0 We prove (a) and (c) and leave proof of the remaining properties for the exercises. Proof (a) Applying the definition of dot product to u · v and v · u, we obtain u · v = U 1 V 1 + U z Vz + . + U n V n = V 1 U 1 + Vz U z + . . + Vn U n = v·u · where the middle equality follows from the fact that multiplication of real numbers is commutative. (c) Using the definitions of scalar multiplication and dot product, we have (cu) · v = [ cu 1 , cu 2 , .

R n , ll u + v ii :s ll u ll + ll v ll Section 1 . 2 Length and Angle: The Dot Product � 23 Proof Since both sides of the inequality are nonnegative, showing that the square of the left-hand side is less than or equal to the square of the right-hand side is equiva lent to proving the theorem. 9 2 2 l l ll ll ll ll ::; u + 2 u · v + v ::; ll n ll 2 + 2 ll n ll ll v ll + ll v ll 2 By Cauchy-Schwarz = ( ll n ll + ll v ll ) 2 as required. Distance The distance between two vectors is the direct analogue of the distance between two points on the real number line or two points in the Cartesian plane.

36. An airplane heading due east has a velocity of 200 miles per hour. A wind is blowing from the north at 40 miles per hour. What is the resultant velocity of the airplane? 37. A boat heads north across a river at a rate of 4 miles per hour. If the current is flowing east at a rate of 3 miles per hour, find the resultant velocity of the boat. 38. Ann is driving a motorboat across a river that is 2 km wide. The boat has a speed of 20 km/h in still water, and the current in the river is flowing at 5 km/h.