vb.net generate qr code How do the polar dot products a b and b a, as defined in Answers 5-5 and 5-6, compare in .NET framework

Creating ANSI/AIM Code 39 in .NET framework How do the polar dot products a b and b a, as defined in Answers 5-5 and 5-6, compare

How do the polar dot products a b and b a, as defined in Answers 5-5 and 5-6, compare
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Answer 5-7
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For any two vectors a and b, the polar dot product is commutative That is a b=b a
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Question 5-8
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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
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Answer 5-8
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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 cross-product 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
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Part One Question 5-9
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What s the right-hand rule for cross products
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Consider again the two vectors c and d that we defined in Question 5-8, and their difference angle qd qc that we defined in Answer 5-8 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 right-hand 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 5-10
How do the polar cross products of two vectors c d and d c compare
Answer 5-10
They have identical magnitudes, but they point in opposite directions
6
Question 6-1
What s the unit imaginary number What s the j operator
Answer 6-1
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 6-2
How is the set of imaginary numbers built up How do we denote such numbers
Answer 6-2
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 real-number coefficient If a 0, then j a = a j = ja We denote negative imaginary numbers as j followed by the absolute value of the real-number coefficient If a < 0, then j ( a) = a j = ja
Review Questions and Answers Question 6-3
How is the set of complex numbers built up How do we denote such numbers
Answer 6-3
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 6-4
How do the complex number 0 + j0, the pure real number 0, and the pure imaginary number j0 compare
Answer 6-4
They are all identical
Question 6-5
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 6-5
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 complex-number ratio, the denominator must not be equal to 0 + j0
Part One Question 6-6
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 6-6
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 6-7
What s the Cartesian complex-number plane What s the polar complex-number plane How are complex vectors defined in these planes
Answer 6-7
Figure 10-4 shows a Cartesian complex-number plane The horizontal axis portrays the realnumber part, and the vertical axis portrays the imaginary-number part A Cartesian complex vector is rendered in standard form, going from the origin to the terminating point corresponding to the complex number Figure 10-5 shows a polar complex-number 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 10-4 are the same size as the radial divisions in Fig 10-5, the vectors in both drawings represent the same complex number
Question 6-8
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