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

1 Answer
Mar 17, 2016

[-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]