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At first glance, this might seem like one of those facts that s intuitively obvious and difficult to prove But all we have to do is work out some arithmetic with fancy characters Let s start with k+(a + b) where k+ is a positive real number, a = (xa,ya), and b = (xb,yb) We can expand the vector sum into an ordered pair, writing the above expression as k+(a + b) = k+[(xa + xb),(ya + yb)] The definition of left-hand scalar multiplication of a Cartesian vector tells us that we can rewrite this as k+(a + b) = {[k+(xa + xb)],[k+(ya + yb)]} In pre-algebra, we learned that real-number multiplication is left-hand distributive over real-number addition, so we can morph the above equation to get k+(a + b) = [(k+xa + k+xb),(k+ya + k+yb)] Let s set this equation aside for a little while We shouldn t forget about it, however, because we re going to come back to it shortly Now, instead of the product of the scalar and the sum of the vectors, let s start with the sum of the scalar products k+a + k+b We can expand the individual vectors into ordered pairs to get k+a + k+b = k+(xa,ya) + k+(xb,yb)
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The definition of left-hand scalar multiplication lets us rewrite this equation as k+a + k+b = (k+xa,k+ya) + (k+xb,k+yb) According to the definition of the Cartesian sum of vectors, we can add the elements of these ordered pairs individually to get a new ordered pair That gives us k+a + k+b = [(k+xa + k+xb),(k+ya + k+yb)] Take a close look at the right-hand side of this equation It s the same as the right-hand side of the equation we put into brain memory a minute ago That equation was k+(a + b) = [(k+xa + k+xb),(k+ya + k+yb)] Taken together, the above two equations show us that k+(a + b) = k+a + k+b
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Mathematicians define two ways in which a vector can be multiplied by another vector The simpler operation is called the dot product and is symbolized by a large dot ( ) Sometimes it s called the scalar product because the end result is a scalar Some texts refer to it as the inner product
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Cartesian dot product Suppose we re given two standard-form vectors a and b in Cartesian coordinates, defined by the ordered pairs
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a = (xa,ya) and b = (xb,yb) The Cartesian dot product a b is the real number we get when we multiply the x values by each other, multiply the y values by each other, and then add the two results The formula is a b = xa xb + ya yb
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An example Consider two standard-form vectors in the Cartesian xy plane, given by the ordered pairs
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a = (4,0)
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80 Vector Multiplication
and b = (3,4) In this case, xa = 4, xb = 3, ya = 0, and yb = 4 We calculate the dot product by plugging the numbers into the formula, getting a b = (4 3) + (0 4) = 12 + 0 = 12
Polar dot product Now let s work in the polar-coordinate plane Imagine two vectors defined by the ordered pairs
a = (qa,ra) and b = (qb,rb) Let q b qa be the angle between vectors a and b, expressed in a rotational sense starting at a and finishing at b as shown in Fig 5-3 We calculate the polar dot product a b by multiplying the magnitude of a by the magnitude of b, and then multiplying that result by the cosine of q b qa to get a b = rarb cos (q b qa)
p /2
(q a, ra)