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The four points lie at the vertices of a parallelogram (believe it or not!)
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Figure 8-6 Vector addition in Cartesian xyz space The terminating points
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of the three vectors a, b, and a + b, along with the origin, lie at the vertices of a parallelogram Perspective distorts the view
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perspective, and from an oblique angle All three vectors project generally in our direction; that is, they re all coming out of the page
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Cartesian negative of a vector To find the Cartesian negative of a standard-form vector in xyz space, we take the negatives of all three coordinate values For example, if we have
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a = (xa,ya,za) then the Cartesian negative vector is a = ( xa, ya, za) As in two-space, the Cartesian negative of a three-space vector always has the same magnitude as the original, but points in the opposite direction
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Cartesian vector difference Let s look again at the two generic vectors
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a = (xa,ya,za)
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136 Vectors in Cartesian Three-Space
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and b = (xb,yb,zb) Suppose we want to subtract b from a We can do this by finding the Cartesian negative of b and then adding that result to a, getting a b = a + ( b) = {[(xa + ( xb)],[(ya + ( yb)],[(za + ( zb)]} = [(xa xb),(ya yb),(za zb)] We can skip the find-the-negative step and simply subtract the coordinate values, but we must be sure to keep the coordinates in the correct order if we do it that way
An example Let s look again at the three standard-form vectors that we worked with a few minutes ago They are
a = (4,0,0) b = (0, 5,0) c = (0,0,3) Suppose we want to find the sum vector a + b We add the x, y, and z coordinates individually to get a + b = (4,0,0) + (0, 5,0) = {(4 + 0),[0 + ( 5)],(0 + 0)} = (4, 5,0) If we add c to the right-hand side of this sum, we get (a + b) + c = (4, 5,0) + (0,0,3) = [(4 + 0),( 5 + 0),(0 + 3)] = (4, 5,3)
Another example Continuing with the same three vectors as previously, let s find the sum b + c We add the x, y, and z coordinates individually to get
b + c = (0, 5,0) + (0,0,3) = [(0 + 0),( 5 + 0),(0 + 3)] = (0, 5,3) Adding a to the left-hand side of this sum, we obtain a + (b + c) = (4,0,0) + (0, 5,3) = {(4 + 0),[0 + ( 5)],(0 + 3)} = (4, 5,3)
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Are you confused
The previous example might lead you to ask, Is vector addition associative in xyz space, just as real-number addition is associative in ordinary algebra The answer is yes The following proof will show you why
Here s a challenge!
Show that if a, b, and c are standard-form vectors in Cartesian xyz space, then addition among them is associative That is (a + b) + c = a + (b + c)
Solution
Let s begin by assigning generic names to the coordinates of each vector Using the same style as we ve been working with all along, we can say that a = (xa,ya,za) b = (xb,yb,zb) c = (xc,yc,zc) When we add a and b using the formula we ve learned, we get a + b = [(xa + xb),(ya + yb),(za + zb)] Adding c to this sum on the right, again using the formula we ve learned, we obtain (a + b) + c = {[(xa + xb) + xc],[(ya + yb) + yc],[(za + zb) + zc]} The associative law for addition of real numbers allows us to regroup each of the three coordinates in the ordered triple to get (a + b) + c = {[xa + (xb + xc)],[ya + (yb + yc)],[za + (zb + zc)]} By definition, we know that {[xa + (xb + xc)],[ya + (yb + yc)],[za + (zb + zc)]} = a + (b + c) By substitution, we have (a + b) + c = a + (b + c)
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