What is the cross product of [-1, -1, 2] and [2, 5, 4] ?
1 Answer
Explanation:
We know that
So 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}))
Another thing that you should know is that cross product is distributive, which means
vecA xx (vecB + vecC) = vecA xx vecB + vecA xx vecC .
We are going to need all of these results for this question.
[-1,-1,2] xx [2,5,4]
= (-hati - hatj + 2hatk) xx (2hati + 5hatj + 4hatk)
= color(white)( (color(black){-hati xx 2hati - hati xx 5hatj - hati xx 4hatk}), (color(black){-hatj xx 2hati - hatj xx 5hatj - hatj xx 4hatk}), (color(black){+2hatk xx 2hati + 2hatk xx 5hatj + 2hatk xx 4hatk}) )
= color(white)( (color(black){-2(vec0) - 5hatk + 4hatj}), (color(black){+2hatk quad - 5(vec0) - 4hati}), (color(black){quad +4hatj quad - 10hati + 8(vec0)}) )
= -14hati + 8hatj + 8hatk
= [-14,8,-3]