Linear Algebra

34 Chapter One. Linear 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 2 .{(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.