What is the cross product of #[-1,0,1]# and #[3, 1, -1] #?

1 Answer
Mar 16, 2016

Answer:

#[-1,2,-1]#

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.

#[-1,0,1] xx [3,1,-1]#

#= (-hati + hatk) xx (3hati + hatj - hatk)#

#= color(white)( (color(black){-hati xx 3hati - hati xx hatj - hati xx (-hatk)}), (color(black){+hatk xx 3hati + hatk xx hatj + hatk xx (-hatk)}) )#

#= color(white)( (color(black){-3(vec0) - hatk - hatj}), (color(black){+ 3hatj qquad - hati - vec0}) )#

#= -hati + 2hatj + -1hatk#

#= [-1,2,-1]#