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What s the Cartesian standard form for a vector in xyz space
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Any vector in xyz space, no matter where its originating and terminating points are located, has an equivalent standard-form vector whose originating point is at (0,0,0) Consider a vector c whose originating point is Q1 and whose terminating point is Q2, such that Q1 = (x1,y1,z1) and Q2 = (x2,y2,z2) The standard form of c , denoted c, has the originating point (0,0,0) and the terminating point Qc such that Qc = (xc,yc,zc) = [(x2 x1),( y2 y1),(z2 z1)] The two vectors c and c have identical direction angles and identical magnitudes That s why we say they re equivalent
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The standard form allows us to uniquely define a vector as an ordered triple that represents the coordinates of its terminating point alone We don t have to worry about the originating point
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Question 8-3
How can we find the magnitude rb of a standard-form vector b in xyz space whose terminating point has the coordinates (xb,yb,zb)
Answer 8-3
We can do it by calculating the distance of the terminating point from the origin In this case, the formula is rb = (xb2 + yb2 + zb2)1/2
Part One Question 8-4
How can we define the direction of a standard-form vector in xyz space whose terminating point has the coordinates (xb,yb,zb)
Answer 8-4
The x, y, and z coordinates implicitly contain all the information we need to define the direction of a standard-form vector in Cartesian three-space But this information is indirect Alternatively, we can define the vector s direction if we know the measures of the angles qx, qy, and qz that the vector subtends relative to the +x, +y, and +z axes, respectively These angles are never negative, and they re never larger than p There is a one-to-one correspondence between all possible vector orientations and all possible values of the ordered triple (qx,qy,qz)
Question 8-5
Imagine two Cartesian xyz space vectors a and b, in standard form with terminating-point coordinates a = (xa,ya,za) and b = (xb,yb,zb) How can we find the sum a + b How can we find the difference a - b How can we find the difference b - a How can we calculate the Cartesian xyz space negative of a vector that s in standard form How does the Cartesian negative compare with the original vector How do the differences a - b and b - a compare
Answer 8-5
We can calculate the sum vector a + b using the formula a + b = [(xa + xb),( ya + yb),(za + zb)] We can calculate the difference vector a b using the formula a - b = [(xa xb),( ya yb),(za zb)] We can find the difference vector b - a using the formula b - a = [(xb xa),( yb ya),(zb za)] To find the Cartesian xyz space negative of a vector that s in standard form, we take the negatives of all three terminating-point coordinate values For example, if we have b = (xb,yb,zb) then its Cartesian negative is -b = ( xb, yb, zb)
Review Questions and Answers
The Cartesian negative has the same magnitude as the original vector, but points in the opposite direction In xyz space, the difference vector b - a is always equal to the Cartesian negative of the difference vector a - b
Question 8-6
What s the left-hand Cartesian product of a scalar and a vector in xyz space What s the righthand Cartesian product of a vector and a scalar in xyz space How do they compare
Answer 8-6
Consider a real-number constant k, along with a standard-form vector a defined in xyz space as a = (xa,ya,za) The left-hand Cartesian product of k and a is ka = (kxa,kya,kza) The right-hand Cartesian product of a and k is ak = (xak,yak,zak) For all real numbers k and all Cartesian xyz space vectors a, we can be sure that ka = ak
Question 8-7
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