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vb.net generate qr code Vector Multiplication in Visual Studio .NET
CHAPTER Recognize USS Code 39 In .NET Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in Visual Studio .NET applications. Code 39 Extended Maker In .NET Using Barcode creation for .NET framework Control to generate, create Code 39 Extended image in Visual Studio .NET applications. Vector Multiplication
Scan ANSI/AIM Code 39 In .NET Framework Using Barcode recognizer for .NET framework Control to read, scan read, scan image in VS .NET applications. Generating Barcode In VS .NET Using Barcode generation for VS .NET Control to generate, create barcode image in Visual Studio .NET applications. We ve seen how vectors add and subtract in two dimensions In this chapter, we ll learn how to multiply a vector by a real number Then we ll explore two different ways in which vectors can be multiplied by each other Reading Barcode In .NET Framework Using Barcode scanner for VS .NET Control to read, scan read, scan image in .NET framework applications. Generate Code 39 Full ASCII In C#.NET Using Barcode generation for .NET Control to generate, create Code39 image in .NET applications. Product of Scalar and Vector
Create Code39 In .NET Framework Using Barcode generator for ASP.NET Control to generate, create Code 3/9 image in ASP.NET applications. Printing Code 3/9 In VB.NET Using Barcode creator for .NET Control to generate, create Code 3/9 image in VS .NET applications. The simplest form of vector multiplication involves changing the magnitude by a realnumber factor called a scalar A scalar is a onedimensional quantity that can be positive, negative, or zero If the scalar is positive, the vector direction stays the same If the scalar is negative, the vector direction reverses If the scalar is zero, the vector disappears Printing Barcode In VS .NET Using Barcode generation for .NET Control to generate, create bar code image in .NET applications. Print Matrix 2D Barcode In Visual Studio .NET Using Barcode printer for Visual Studio .NET Control to generate, create 2D Barcode image in Visual Studio .NET applications. Cartesian vector times positive scalar Imagine a standardform vector a in the Cartesian xy plane, defined by an ordered pair whose coordinates are xa and ya, so that Bar Code Creation In Visual Studio .NET Using Barcode printer for .NET Control to generate, create bar code image in VS .NET applications. Intelligent Mail Generation In .NET Framework Using Barcode maker for .NET framework Control to generate, create USPS Intelligent Mail image in .NET framework applications. a = (xa,ya) Suppose that we multiply a positive scalar k+ by each of the vector coordinates individually, getting two new coordinates Mathematically, we write this as k+a = (k+ xa,k+ ya) This vector is called the lefthand Cartesian product of k+ and a If we multiply both original coordinates on the right by k+ instead, we get a k+ = (xak+,yak+) Code 3/9 Creation In None Using Barcode generator for Microsoft Word Control to generate, create Code 39 Extended image in Office Word applications. Recognize GS1  12 In Visual C# Using Barcode reader for .NET framework Control to read, scan read, scan image in VS .NET applications. Vector Multiplication
UPCA Reader In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. Make Barcode In ObjectiveC Using Barcode printer for iPhone Control to generate, create barcode image in iPhone applications. That s the righthand Cartesian product of a and k+ The individual coordinates of k+a and ak+ are products of real numbers We learned in prealgebra that realnumber multiplication is commutative, so it follows that k+a = (k+ xa,k+ya) = (xak+,yak+) = a k+ We ve just shown that multiplication of a Cartesianplane vector by a positive scalar is commutative We don t have to worry about whether we multiply on the left or the right; we can simply talk about the Cartesian product of the vector and the positive scalar Recognizing Bar Code In Visual C# Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET applications. Barcode Decoder In Java Using Barcode decoder for Java Control to read, scan read, scan image in Java applications. An example Figure 51 illustrates the Cartesian vector ( 1, 2) as a solid, arrowed line segment If we multiply this vector by 3 on the left, we get Recognizing Code 3/9 In Visual C#.NET Using Barcode decoder for .NET framework Control to read, scan read, scan image in Visual Studio .NET applications. Barcode Recognizer In VB.NET Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET framework applications. 3 ( 1, 2) = {[3 ( 1)],[3 ( 2)]} = ( 3, 6) If we multiply the original vector by 3 on the right, we get ( 1, 2) 3 = [( 1 3)],( 2 3)] = ( 3, 6) The new vector is shown as a dashed, gray, arrowed line segment pointing in the same direction as the original vector, but 3 times as long Figure 51 Cartesian products of the scalars 3 and 3
with the vector ( 1, 2) Product of Scalar and Vector
Cartesian vector times negative scalar Now suppose we want to multiply a by a negative scalar instead of a positive scalar Let s call the scalar k The lefthand Cartesian product of k and a is k a = (k xa,k ya) The righthand Cartesian product is a k = (xak ,yak ) As with the positive constant, the commutative property of realnumber multiplication tells us that k a = (k xa,k ya) = (xak ,yak ) = ak We don t have to worry about whether we multiply on the left or the right We get the same result either way An example Once again, look at Fig 51 with the vector ( 1, 2) shown as a solid, arrowed line segment When we multiply it by the scalar 3 on the left, we obtain 3 ( 1, 2) = {[ 3 ( 1)],[ 3 ( 2)]} = (3,6) Multiplying by the scalar on the right, we get ( 1, 2) ( 3) = {[ 1 ( 3)],[ 2 ( 3)]} = (3,6) This result is shown as a dashed, gray, arrowed line segment pointing in the opposite direction from the original vector, and 3 times as long Polar vector times positive scalar Imagine some vector a in the polarcoordinate plane whose direction angle is qa and whose magnitude is ra If it s in standard form, we can express it as the ordered pair a = (qa,ra) When we multiply a on the left by a positive scalar k+, the angle remains the same, but the magnitude becomes k+ra This gives us the lefthand polar product of k+ and a, which is k+a = (qa,k+ra) If we multiply a on the right by k+, we get the righthand polar product of a and k+, which is a k+ = (qa,rak+)

