What is the cross product of $⟨ - 3, - 6, - 3 ⟩$ $<< -3,-6,-3 >>$ and $⟨ - 5, 2, - 7 ⟩$ $<< -5,2,-7 >>$ ?

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Answer 1 · 2017-10-01T05:37:27

The vector is $= ⟨ 48, - 6, - 36 ⟩$ $=〈48,-6,-36〉$

The cross product of 2 vectors is calculated with the determinant

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

where $〈d,e,f〉$ and $〈g,h,i〉$ are the 2 vectors

Here, we have $veca=〈-3,-6,-3〉$ and $vecb=〈-5,2,-7〉$

Therefore,

$| (veci,vecj,veck), (-3,-6,-3), (-5,2,-7) |$

$=veci| (-6,-3), (2,-7) | -vecj| (-3,-3), (-5,-7) | +veck| (-3,-6), (-5,2) |$

$=veci((-6*-7)-(2*-3))-vecj((-3*-7)-(-5*-3))+veck((-3*2)-(-6*-5))$

$=〈48,-6,-36〉=vecc$

Verification by doing 2 dot products (inner products)

$〈48,-6,-36〉.〈-3,-6,-3〉=-48*3+6*6+36*3=0$

$〈48,-6,-36〉.〈-5,2,-7〉=-48*5-6*2+36*7=0$

So,

$vecc$ is perpendicular to both $veca$ and $vecb$

What is the cross product of << -3,-6,-3 >>⟨−3,−6,−3⟩<< -3,-6,-3 >> and << -5,2,-7 >>⟨−5,2,−7⟩<< -5,2,-7 >>?