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Figure 5-3 To find the polar dot product of two
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vectors, we must know the angle between them as we rotate from the first vector (in this case a) to the second vector (in this case b)
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An example Suppose that we re given two vectors a and b in the polar plane, and told that their coordinates are
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a = (p /6,3) b = (5p /6,2) In this situation, ra = 3, rb = 2, qa = p /6, and qb = 5p /6 We have qb qa = 5p /6 p /6 = 2p /3 Therefore, the dot product is a b = rarb cos (qb qa) = 3 2 cos (2p /3) = 3 2 ( 1/2) = 3
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Do you wonder if the dot product of two polar-plane vectors is always equal to the dot product of the same vectors in the Cartesian plane when expressed in standard form The answer is yes Let s find out why
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Here s a challenge!
Prove that for any two vectors a and b in two-space, the polar dot product a b is the same as the Cartesian dot product a b when both vectors are in standard form
Solution
We will start with the polar versions of the vectors, calling them a = (qa,ra) and b = (qb,rb) Let s convert these vectors to Cartesian form We can use the formulas for conversion of points from polar to Cartesian coordinates (from Chap 3) When we apply them to vector a, we get xa = ra cos qa and ya = ra sin qa so the standard Cartesian form of the vector is
a = [(ra cos qa),(ra sin qa)]
82 Vector Multiplication
When we apply the same conversion formulas to b, we obtain xb = rb cos qb and yb = rb sin qb so the standard Cartesian form is b = [(rb cos qb),(rb sin qb)] The Cartesian dot product of the two vectors is a b = xa xb + ya yb Substituting the values we found for the individual vector coordinates, we get a b = (ra cos qa)(rb cos qb) + (ra sin qa)(rb sin qb) = rarb (cos qa cos qb + sin qa sin qb) As we think back to our trigonometry courses, we recall that there s a trigonometric identity telling us how to expand the cosine of the difference between two angles When we name the angles so they apply to our situation here, that formula becomes cos (qb qa) = cos qa cos qb + sin qa sin qb We can substitute the left-hand side of this identity in the last part of the long equation we got a minute ago for the dot product, obtaining a b = rarb cos (qb qa) This is the formula for the polar dot product! We ve taken the polar versions of a and b, found their Cartesian dot product, and then found that it s identical to the polar dot product We can now say, Quod erat demonstradum That s Latin for Which was to be proved Some mathematicians write the abbreviation for this expression, QED, when they ve finished a proof
Cross Product of Two Vectors
The more complicated (and interesting) way to multiply two vectors by each other gives us a third vector that jumps out of the coordinate plane This operation is known as the cross product Some mathematicians call it the vector product The cross product of two vectors a and b is written as a b
Cross Product of Two Vectors
Polar cross product Imagine two arbitrary vectors in the polar-coordinate plane, expressed in standard form as ordered pairs
a = (qa,ra) and b = (qb,rb) The magnitude of a b is always nonnegative by default, and is easy to define When a and b are in standard form, the originating point of a b is at the coordinate origin, so all three vectors start at the same spot The direction of a b is always along the line passing through the origin at a right angle to the plane containing a and b But it s quite a trick to figure out in which direction the cross vector product points along this line! Suppose that the difference qb qa between the direction angles is positive but less than p, as shown in the example of Fig 5-4 If we start at vector a and rotate until we get to vector b, we turn through an angle of qb qa To calculate the magnitude of a b (which we will denote
(q b, rb)
p /2
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