What is the cross product of #[-1, -1, 2]# and #[-1, 2, 2] #?
1 Answer
Explanation:
The cross product between two vectors
where
For of the unit vectors
#color(white)( (color(black){hati xx hati = vec0}, color(black){qquad hati xx hatj = hatk}, color(black){qquad hati xx hatk = -hatj}), (color(black){hatj xx hati = -hatk}, color(black){qquad hatj xx hatj = vec0}, color(black){qquad hatj xx hatk = hati}), (color(black){hatk xx hati = hatj}, color(black){qquad hatk xx hatj = -hati}, color(black){qquad hatk xx hatk = vec0}))#
Also, note that cross product is distributive.
#vecA xx (vecB + vecC) = vecA xx vecB + vecA xx vecC# .
So for this question.
#[-1,-1,2] xx [-1,2,2]#
#= (-hati - hatj + 2hatk) xx (-hati + 2hatj + 2hatk)#
#= color(white)( (color(black){-hati xx (-hati) - hati xx 2hatj - hati xx 2hatk}), (color(black){-hatj xx (-hati) - hatj xx 2hatj - hatj xx 2hatk}), (color(black){+2hatk xx (-hati) + 2hatk xx 2hatj + 2hatk xx 2hatk}) )#
#= color(white)( (color(black){vec0 - 2hatk quad qquad + 2hatj}), (color(black){-hatk - 2(vec0) - 2hati}), (color(black){- 2hatj - 4hati quad - 4(vec0)}) )#
#= -6hati - 3hatk#
#= [-6,0,-3]#