What is the cross product of #<4 , 5 ,-7 ># and #<5 ,1 ,-3 >#?

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Answer 1 · 2018-05-18T17:14:08

The vector is #=〈-8,-23,-21〉#

The cross product of 2 vectors is calculated with the determinant

#| (veci,vecj,veck), (d,e,f), (g,h,i) | #

where #veca=〈d,e,f〉# and #vecb=〈g,h,i〉# are the 2 vectors

Here, we have #veca=〈4,5,-7〉# and #vecb=〈5,1,-3〉#

Therefore,

#| (veci,vecj,veck), (4,5,-7), (5,1,-3) | #

#=veci| (5,-7), (1,-3) | -vecj| (4,-7), (5,-3) | +veck| (4,5), (5,1) | #

#=veci((5)*(-3)-(-7)*(1))-vecj((4)*(-3)-(-7)*(5))+veck((4)*(1)-(5)*(5))#

#=〈-8,-23,-21〉=vecc#

Verification by doing 2 dot products

#〈-8,-23,-21〉.〈4,5,-7〉=(-8)*(4)+(-23)*(5)+(-21)*(-7)=0#

#〈-8,-23,-21〉.〈5,1,-3〉=(-8)*(5)+(-23)*(1)+(-21)*(-3)=0#

So,

#vecc# is perpendicular to #veca# and #vecb#