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34 Chapter One. <strong>Linear</strong> SystemsWe often draw the arrow as starting at the origin, and we then say it is in thecanonical position (or natural position). When the vector(b1 − a 1b 2 − a 2)is in its canonical position then it extends to the endpoint (b 1 − a 1 , b 2 − a 2 ).We typically just refer to “the point(12)”rather than “the endpoint of the canonical position of” that vector. Thus, wewill call both of these sets R n .{(x 1 , x 2 ) ∣ ∣ x 1 , x 2 ∈ R} {(x1x 2) ∣∣x 1 , x 2 ∈ R}In the prior section we defined vectors and vector operations with an algebraicmotivation;( ) ( ) ( ) ( ) ( )v1 rv1 v1 w1 v1 + wr · =+ =1v 2 rv 2 v 2 w 2 v 2 + w 2we can now interpret those operations geometrically. For instance, if ⃗v representsa displacement then 3⃗v represents a displacement in the same directionbut three times as far, and −1⃗v represents a displacement of the same distanceas ⃗v but in the opposite direction.−⃗v⃗v3⃗vAnd, where ⃗v and ⃗w represent displacements, ⃗v + ⃗w represents those displacementscombined.⃗v + ⃗w⃗w⃗vThe long arrow is the combined displacement in this sense: if, in one minute, aship’s motion gives it the displacement relative to the earth of ⃗v and a passenger’smotion gives a displacement relative to the ship’s deck of ⃗w, then ⃗v + ⃗w isthe displacement of the passenger relative to the earth.Another way to understand the vector sum is with the parallelogram rule.Draw the parallelogram formed by the vectors ⃗v 1 , ⃗v 2 and then the sum ⃗v 1 + ⃗v 2extends along the diagonal to the far corner.
Section II. <strong>Linear</strong> Geometry of n-Space 35⃗w⃗v + ⃗wThe above drawings show how vectors and vector operations behave in R 2 .We can extend to R 3 , or to even higher-dimensional spaces where we have nopictures, with the obvious generalization: the free vector that, if it starts at(a 1 , . . . , a n ), ends at (b 1 , . . . , b n ), is represented by this column⎛ ⎞b 1 − a 1⎜ ⎟⎝ . ⎠b n − a n(vectors are equal if they have the same representation), we aren’t too carefulto distinguish between a point and the vector whose canonical representationends at that point,⎛⎞v 1R n ⎜= { .⎝ .v n⃗v⎟⎠ ∣ ∣ v 1 , . . . , v n ∈ R}and addition and scalar multiplication are component-wise.Having considered points, we now turn to the lines. In R 2 , the line through(1, 2) and (3, 1) is comprised of (the endpoints of) the vectors in this set( ( ) 1 2 ∣∣{ + t · t ∈ R}2)−1That description expresses this picture.(2−1)=(31)−(12)The vector associated with the parameter t has its whole body in the line — itis a direction vector for the line. Note that points on the line to the left of x = 1are described using negative values of t.In R 3 , the line through (1, 2, 1) and (2, 3, 2) is the set of (endpoints of)vectors of this form) ) ∣∣t ∈ R}{( 121+ t ·( 111
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