What is the cross product of #[0,8,5]# and #[1,2,-4] #?

1 Answer
Apr 1, 2016

Answer:

#[0,8,5] xx [1,2,-4] = [-42,5,-8]#

Explanation:

The cross product of #vecA# and #vecB# is given by

#vecA xx vecB = ||vecA|| * ||vecB|| * sin(theta) hatn#,

where #theta# is the positive angle between #vecA# and #vecB#, and #hatn# is a unit vector with direction given by the right hand rule.

For the unit vectors #hati#, #hatj# and #hatk# in the directions of #x#, #y# and #z# respectively,

#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, cross product is distributive, which means

#vecA xx (vecB + vecC) = vecA xx vecB + vecA xx vecC#.

For this question,

#[0,8,5] xx [1,2,-4]#

#= (8hatj + 5hatk) xx (hati + 2hatj - 4hatk)#

#= color(white)( (color(black){qquad 8hatj xx hati + 8hatj xx 2hatj + 8hatj xx (-4hatk)}), (color(black){+5hatk xx hati + 5hatk xx 2hatj + 5hatk xx (-4hatk)}) )#

#= color(white)( (color(black){-8hatk + 16(vec0) - 32hati}), (color(black){qquad +5hatj - quad 10hati quad - 20(vec0)}) )#

#= -42hati + 5hatj - 8hatk#

#= [-42,5,-8]#