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5 Code 39 Extended Recognizer In Visual Studio .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Code 3/9 Encoder In .NET Framework Using Barcode encoder for VS .NET Control to generate, create USS Code 39 image in Visual Studio .NET applications. 1 We ve been given the Cartesian vectors a = (5, 5) and b = ( 5,5) Code39 Scanner In VS .NET Using Barcode scanner for VS .NET Control to read, scan read, scan image in Visual Studio .NET applications. Bar Code Printer In VS .NET Using Barcode generator for VS .NET Control to generate, create bar code image in VS .NET applications. 490 WorkedOut Solutions to Exercises: 19 Read Bar Code In .NET Framework Using Barcode decoder for VS .NET Control to read, scan read, scan image in VS .NET applications. Print ANSI/AIM Code 39 In C#.NET Using Barcode printer for VS .NET Control to generate, create USS Code 39 image in VS .NET applications. When we multiply a on the left by 4, we get 4a = 4 (5, 5) = {(4 5), [4 ( 5)]} = (20, 20) When we multiply b on the left by 4, we get 4b = 4 ( 5,5) = {[ 4 ( 5)],( 4 5)} = (20, 20) 2 The Cartesian vector a has the coordinates xa = 5 and ya = 5, so it terminates in the fourth quadrant The direction angle for the polar form of a can be found using the conversion formula for a vector in the fourth quadrant, giving us qa = 2p + Arctan ( 5/5) = 2p + Arctan ( 1) = 2p + ( p /4) = 7p /4 The magnitude of a is found by the distance formula ra = [52 + ( 5)2]1/2 = (25 + 25)1/2 = 501/2 Therefore, the polar version of a is a = (7p /4,501/2) The Cartesian version of b has xb = 5 and yb = 5 It terminates in the second quadrant Using the conversion formula for the direction angle of a vector in that quadrant, we get qb = p + Arctan [5/( 5)] = p + Arctan ( 1) = p + ( p /4) = 3p /4 The magnitude of b is rb = [( 5)2 + 52]1/2 = (25 + 25)1/2 = 501/2 Therefore, the polar version of b is b = (3p /4,501/2) When we multiply a on the left by 4, we get 4a = 4 (7p /4,501/2) = (7p /4,8001/2) When we multiply b on the left by 4, we get 4b = 4 (3p /4,501/2) = (3p /4, 8001/2) = (7p /4,8001/2) Code39 Maker In Visual Studio .NET Using Barcode creation for ASP.NET Control to generate, create Code 39 Full ASCII image in ASP.NET applications. Code39 Drawer In Visual Basic .NET Using Barcode generation for .NET Control to generate, create Code 39 image in .NET applications. 5
Barcode Generation In Visual Studio .NET Using Barcode creation for .NET Control to generate, create barcode image in .NET framework applications. Barcode Creator In VS .NET Using Barcode maker for .NET Control to generate, create barcode image in .NET framework applications. In the last step in the equation for 4b, we must take the absolute value of the negative magnitude coordinate, because we can t allow a vector to have negative magnitude We do this by reversing the direction, in this case by adding p to the angle 3 We want to prove that positivescalar multiplication is righthand distributive over vector subtraction in the Cartesian xy plane Let s start with (a b)k+ where a = (xa,ya), b = (xb,yb), and k+ is a positive real number Expanding the vectors into their ordered pairs in our initial expression, we get (a b)k+ = [(xa xb),( ya yb)]k+ The definition of righthand scalar multiplication tells us that we can morph this equation to obtain (a b)k+ = {[(xa xb)k+],[( ya yb)]k+} The righthand distributive law for real numbers allows us to transform the equation further, getting (a b)k+ = [(xak+ xbk+),( yak+ ybk+)] Let s put this equation aside for moment We ll come back to it! Now, instead of the product of the vector difference and the constant, let s start with the difference between the products ak+ bk+ We can expand the individual vectors into ordered pairs to get ak+ bk+ = (xa,ya)k+ (xb,yb)k+ By the definition of righthand scalar multiplication, we have ak+ bk+ = (xak+,yak+) (xbk+,ybk+) When we add the elements of the ordered pairs individually to get a new ordered pair, we obtain ak+ bk+ = [(xak+ xbk+),( yak+ ybk+)] The righthand side of this equation is the same as the righthand side of the equation we put aside a minute ago That equation, once again, is (a b)k+ = [(xak+ xbk+),( yak+ ybk+)] Code39 Drawer In .NET Using Barcode generator for Visual Studio .NET Control to generate, create Code 3 of 9 image in Visual Studio .NET applications. Encode MSI Plessey In .NET Framework Using Barcode creator for VS .NET Control to generate, create MSI Plessey image in .NET applications. 492 WorkedOut Solutions to Exercises: 19 Generating Code 128 Code Set A In Java Using Barcode creator for BIRT reports Control to generate, create USS Code 128 image in Eclipse BIRT applications. Scanning ECC200 In Visual Basic .NET Using Barcode reader for .NET framework Control to read, scan read, scan image in .NET framework applications. Taken together, the above two equations show us that (a b)k+ = ak+ bk+ 4 We ve been given the Cartesian vectors a = (4,4) and b = ( 7,7) We can define the coordinate values as xa = 4, xb = 7, ya = 4, and yb = 7 The Cartesian dot product of a and b, in that order, is therefore a b = xaxb + yayb = 4 ( 7) + 4 7 = 28 + 28 = 0 The Cartesian dot product of b and a, in that order, is b a = xbxa + ybya = 7 4 + 7 4 = 28 + 28 = 0 5 The Cartesian vector a has the coordinates xa = 4 and ya = 4, so it terminates in the first quadrant The direction angle for the polar form of a is therefore qa = Arctan (4/4) = Arctan 1 = p /4 The magnitude of a is ra = [42 + 42]1/2 = (16 + 16)1/2 = 321/2 so the polar form of a is a = (p /4,321/2) The Cartesian vector b has xb = 7 and yb = 7 It terminates in the second quadrant Using the conversion formula for the direction angle of a vector in the second quadrant, we get qb = p + Arctan [7/( 7)] = p + Arctan ( 1) = p + ( p /4) = 3p /4 The magnitude of b is rb = [( 7)2 + 72]1/2 = (49 + 49)1/2 = 981/2 Therefore, the polar version of b is b = (3p /4,981/2) Code 39 Extended Maker In Java Using Barcode encoder for Java Control to generate, create ANSI/AIM Code 39 image in Java applications. Reading Data Matrix ECC200 In Visual C#.NET Using Barcode recognizer for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. Generate Barcode In VB.NET Using Barcode creation for VS .NET Control to generate, create bar code image in Visual Studio .NET applications. Barcode Creation In Visual C# Using Barcode creation for Visual Studio .NET Control to generate, create bar code image in .NET framework applications. Encoding Code 39 In C#.NET Using Barcode generation for VS .NET Control to generate, create Code 39 Extended image in VS .NET applications. Barcode Generator In None Using Barcode maker for Online Control to generate, create barcode image in Online applications. 
