vb.net generate qr code What restrictions apply when we work with vectors in the polar-coordinate plane in VS .NET

Make USS Code 39 in VS .NET What restrictions apply when we work with vectors in the polar-coordinate plane

What restrictions apply when we work with vectors in the polar-coordinate plane
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A polar vector is not allowed to have a negative radius, a negative direction angle, or a direction angle of 2p or more These constraints prevent ambiguities, so we can be confident that the set of all polar-plane vectors can be paired off in a one-to-one correspondence with the set of all Cartesian-plane vectors
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Suppose we re given two vectors in polar coordinates What s the best way to find their sum and difference What s the best way to find the negative of a vector in polar coordinates
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The best way to add or subtract polar vectors is to convert them to Cartesian vectors in standard form, then add or subtract those vectors, and finally convert the result back to polar form The best way to find the negative of a polar vector is to reverse its direction and leave the magnitude the same Suppose we have a = (qa,ra) If 0 qa < p, then the polar negative is -a = [(qa + p ),ra] If p qa < 2p, then the polar negative is -a = [(qa p ),ra]
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Question 5-1
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What s the left-hand Cartesian product of a scalar and a vector What s the right-hand Cartesian product of a vector and a scalar How do they compare
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Answer 5-1
Consider a real-number constant k, along with a standard-form vector a defined in the xy plane as a = (xa,ya) The left-hand Cartesian product of k and a is ka = (kxa,kya) The right-hand Cartesian product of a and k is ak = (xak,yak) The left- and right-hand products of a scalar and a Cartesian vector are always the same For all real numbers k and all Cartesian vectors a, we can be sure that ka = ak
Question 5-2
What s the left-hand polar product of a positive scalar and a vector What s the right-hand polar product of a vector and a positive scalar How do they compare
Review Questions and Answers Answer 5-2
Imagine a polar vector a with angle qa and radius ra, such that a = (qa,ra) When we multiply a on the left by a positive scalar k+, we get k+a = (qa,k+ra) When we multiply a on the right by k+, we get ak+ = (qa,rak+) The left- and right-hand polar products of a positive scalar and a polar vector are always the same For all positive real numbers k+ and all polar vectors a, we can be sure that k+a = ak+
Question 5-3
What s the left-hand polar product of a negative scalar and a vector What s the right-hand polar product of a vector and a negative scalar How do they compare
Answer 5-3
Once again, suppose we have a polar vector a with angle qa and radius ra, such that a = (qa,ra) When we multiply a on the left by a negative scalar k , we get k a = [(qa + p ),( k ra)] if 0 qa < p, and k a = [(qa p ),( k ra)] if p qa < 2p Because k is negative, k is positive, so k ra is positive, ensuring that we get a positive radius for the resultant vector If we multiply a on the right by k , we get ak = [(qa + p ),ra( k )] if 0 qa < p, and ak = [(qa p ),ra( k )]
Part One
if p qa < 2p Because k is negative, k is positive, so ra( k ) is positive, ensuring that we get a positive radius for the resultant vector For all negative real numbers k and all polar vectors a, k a = ak
Question 5-4
Suppose we re given two standard-form vectors a and b, defined by the ordered pairs a = (xa,ya) and b = (xb,yb) What s the Cartesian dot product a b What s the Cartesian dot product b a How do they compare
Answer 5-4
The Cartesian dot product a b is a real number given by a b = xaxb + yayb and the Cartesian dot product b a is a real number given by b a = xbxa + ybya The Cartesian dot product is commutative, so for any two vectors a and b in the xy plane, we can be confident that a b=b a
Question 5-5
Imagine a polar vector a with angle qa and radius ra, such that a = (qa,ra) and a polar vector b with angle qb and radius rb, such that b = (qb,rb) What s the polar dot product a b
Answer 5-5
Let qb qa be the angle as we rotate from a to b The polar dot product a b is given by the formula a b = rarb cos (qb qa)
Review Questions and Answers Question 5-6
Consider the same two polar vectors as we worked with in Question and Answer 5-5 What s the polar dot product b a
Answer 5-6
We can define this dot product by reversing the roles of the vectors in the previous problem Let qa qb be the angle going from b to a The polar dot product b a is given by the formula b a = rbra cos (qa qb)
Question 5-7
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