What is the cross product of #[-1, -1, 2]# and #[2, 5, 4] #?

1 Answer
Mar 17, 2016

Answer:

#[-14,8,-3]#

Explanation:

We know that #vecA xx vecB = ||vecA|| * ||vecB|| * sin(theta) hatn#, where #hatn# is a unit vector given by the right hand rule.

So for of the unit vectors #hati#, #hatj# and #hatk# in the direction of #x#, #y# and #z# respectively, we can arrive at the following results.

#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]#