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

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Answer 1 · 2018-05-18T17:14:08

The vector is $= ⟨ - 8, - 23, - 21 ⟩$ $=〈-8,-23,-21〉$

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 $vecb$

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