88CHAPTER 1. COORDINATES AND VECTORSSimilarly,a1−→ı= 2vectorA(△OP1Q1)=−→ıvextendsinglevextendsinglevextendsinglevextendsingley1z1y2z2vextendsinglevextendsinglevextendsinglevextendsingle.Finally, noting that the direction from which thepositivez-axis iscounterclockwisefrom the positivex-axis is−−→, we havea2−→= 2vectorA(△OP2Q2)=−−→vextendsinglevextendsinglevextendsinglevextendsinglex1z1x2z2vextendsinglevextendsinglevextendsinglevextendsingle.Adding these yields the desired formula.In each projection, we used the 2×2 determinant obtained by omitting thecoordinate along whose axis we were projecting. The resulting formula canbe summarized in terms of the array of coordinates of−→vand−→wparenleftbiggx1y1z1x2y2z2parenrightbiggby saying: the coefficient of the standard basis vector in a given coordinatedirection is the 2×2 determinant obtained by eliminating thecorresponding column from the above array, and multiplying by
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