 Home
 Products
 Integration
 Tutorial
 Barcode FAQ
 Purchase
 Company
vb.net generate qr code How do the polar dot products a b and b a, as defined in Answers 55 and 56, compare in .NET framework
How do the polar dot products a b and b a, as defined in Answers 55 and 56, compare Reading USS Code 39 In VS .NET Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET framework applications. Make Code39 In .NET Framework Using Barcode printer for Visual Studio .NET Control to generate, create Code39 image in VS .NET applications. Answer 57 Reading Code39 In Visual Studio .NET Using Barcode reader for VS .NET Control to read, scan read, scan image in .NET applications. Encoding Barcode In VS .NET Using Barcode creation for .NET framework Control to generate, create bar code image in .NET framework applications. For any two vectors a and b, the polar dot product is commutative That is a b=b a
Barcode Scanner In Visual Studio .NET Using Barcode decoder for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications. Drawing Code 3/9 In Visual C#.NET Using Barcode creator for VS .NET Control to generate, create Code 39 Full ASCII image in .NET applications. Question 58 Code 39 Extended Encoder In VS .NET Using Barcode maker for ASP.NET Control to generate, create ANSI/AIM Code 39 image in ASP.NET applications. Encoding ANSI/AIM Code 39 In VB.NET Using Barcode creator for Visual Studio .NET Control to generate, create Code 39 Extended image in .NET framework applications. Imagine a polar vector c with angle qc and radius rc, such that c = (qc,rc) and a polar vector d with angle qd and radius rd, such that d = (qd,rd ) What s the polar cross product c d Paint Linear Barcode In Visual Studio .NET Using Barcode creator for VS .NET Control to generate, create Linear image in .NET applications. GS1 DataBar Stacked Drawer In .NET Framework Using Barcode generator for Visual Studio .NET Control to generate, create DataBar image in Visual Studio .NET applications. Answer 58 Code 128 Code Set C Generator In VS .NET Using Barcode maker for VS .NET Control to generate, create Code 128C image in .NET framework applications. Identcode Maker In .NET Framework Using Barcode drawer for VS .NET Control to generate, create Identcode image in Visual Studio .NET applications. Imagine that we start at vector c and rotate counterclockwise until we get to vector d, so we turn through an angle of qd qc Suppose that 0 < qd qc < p To calculate the magnitude rc d of the crossproduct vector c d, we use the formula rc d = rcrd sin (qd qc) In this situation, c d points toward us If p < qd qc < 2p, we can consider the difference angle to be 2p + qc qd Then the magnitude of c d is rc d = rcrd sin (2p + qc qd ) and it points away from us Bar Code Generation In Visual Basic .NET Using Barcode printer for .NET Control to generate, create barcode image in Visual Studio .NET applications. Print UCC  12 In Visual C#.NET Using Barcode maker for VS .NET Control to generate, create UCC  12 image in .NET framework applications. Part One Question 59 Matrix 2D Barcode Generator In .NET Using Barcode maker for ASP.NET Control to generate, create 2D Barcode image in ASP.NET applications. Barcode Generation In None Using Barcode encoder for Online Control to generate, create bar code image in Online applications. What s the righthand rule for cross products
Code39 Decoder In Java Using Barcode recognizer for Java Control to read, scan read, scan image in Java applications. Create Barcode In None Using Barcode maker for Software Control to generate, create barcode image in Software applications. Answer 59 Code 3 Of 9 Drawer In Java Using Barcode creator for BIRT Control to generate, create ANSI/AIM Code 39 image in BIRT applications. GTIN  12 Maker In None Using Barcode creator for Font Control to generate, create GS1  12 image in Font applications. Consider again the two vectors c and d that we defined in Question 58, and their difference angle qd qc that we defined in Answer 58 If 0 < qd qc < p, point your right thumb out, and curl your fingers counterclockwise from c to d If p < qd qc < 2p, point your right thumb out, and curl your righthand fingers clockwise from c to d Your thumb will then point in the general direction of c d The vector c d is always perpendicular to the plane defined by c and d Question 510 How do the polar cross products of two vectors c d and d c compare
Answer 510 They have identical magnitudes, but they point in opposite directions
6
Question 61 What s the unit imaginary number What s the j operator
Answer 61 These expressions both refer to the positive square root of 1 If we denote it as j, then j = ( 1)1/2 and j 2 = 1 Question 62 How is the set of imaginary numbers built up How do we denote such numbers
Answer 62 If we multiply j by a nonnegative real number a, we get a nonnegative imaginary number If we multiply j by a negative real number a, we get a negative imaginary number We denote nonnegative imaginary numbers by writing j followed by the realnumber coefficient If a 0, then j a = a j = ja We denote negative imaginary numbers as j followed by the absolute value of the realnumber coefficient If a < 0, then j ( a) = a j = ja Review Questions and Answers Question 63 How is the set of complex numbers built up How do we denote such numbers
Answer 63 A complex number is the sum of a real number and an imaginary number If a is a real number and b is a nonnegative real number, then the general form for a complex number is a + jb If a is a real number and b is a negative real number, then we have a + j( b) but it s customary to write the absolute value of b after j, and use a minus sign instead of a plus sign in the expression That gives us the general form a jb Question 64 How do the complex number 0 + j0, the pure real number 0, and the pure imaginary number j0 compare
Answer 64 They are all identical
Question 65 How do we find the sum of two complex numbers a + jb and c + jd How do we find their difference How do we find their product How do we find their ratio Answer 65 When we want to add, we use the formula (a + jb) + (c + jd ) = (a + c) + j(b + d ) When we want to subtract, we use the formula (a + jb) (c + jd ) = (a c) + j(b d ) When we want to multiply, we use the formula (a + jb)(c + jd ) = (ac bd ) + j(ad + bc) When we want to find the ratio, we use the formula (a + jb) / (c + jd ) = [(ac + bd ) / (c2 + d 2)] + j [(bc ad ) / (c2 + d 2)] In a complexnumber ratio, the denominator must not be equal to 0 + j0 Part One Question 66 What are complex conjugates What happens when we add a complex number to its conjugate What happens when we multiply a complex number by its conjugate Answer 66 Complex conjugates have identical coefficients, but opposite signs between the real and imaginary parts, as in a + jb and a jb When we add a complex number to its conjugate, we get (a + jb) + (a jb) = 2a When we multiply a complex number by its conjugate, we get (a + jb)(a jb) = a2 + b2 Question 67 What s the Cartesian complexnumber plane What s the polar complexnumber plane How are complex vectors defined in these planes Answer 67 Figure 104 shows a Cartesian complexnumber plane The horizontal axis portrays the realnumber part, and the vertical axis portrays the imaginarynumber part A Cartesian complex vector is rendered in standard form, going from the origin to the terminating point corresponding to the complex number Figure 105 shows a polar complexnumber plane Polar complex vectors are defined in terms of their direction angle and magnitude, instead of their real and imaginary parts Assuming that the axis divisions in Fig 104 are the same size as the radial divisions in Fig 105, the vectors in both drawings represent the same complex number Question 68

