What is the cross product of #[3,2, 5]# and #[2, -5, 8] #?

1 Answer
Mar 12, 2018

Answer:

By hand and then checked with MATLAB: [41 -14 -19]

Explanation:

When you take a cross product, I feel like it makes things easier to add in the unit vector directions #[ hat i hat j hat k ] # which are in the x, y, and z directions respectively.

We'll use all three since these are 3-D vectors that we're dealing with. If it was 2d you'd only have to use #hati# and #hatj#

Now we set up a 3x3 matrix as follows (Socratic doesn't give me a good way to do multidimensional matrices, sorry!):

#|hati hatj hatk|#
#|3 2 5|#
#|2 -5 8|#

Now, starting at each unit vector, go diagonal from left to right, taking the product of those numbers:

#(2*8)hati (5*2)hatj (3*-5)hatk#

#=16hati 10hatj -15hatk#

Next, take the products of the values going from right to left; again, starting at the unit vector:

#(5*-5)hati (3*8)hatj (2*2)hatk#

#=-25hati 24hatj 4hatk#

Finally, take the first set and subtract the second set from it

#[16hati 10hatj -15hatk]-[-25hati 24hatj 4hatk]#
#=(16-(-25))hati (10-24)hatj (-15-4)hatk#
#=41hati -14hatj -19hatk#

this can now be re-written in matrix form, with #hati#, #hatj#, and #hatk# removed since it's staying a 3-D vector:

#color(red)("[41 -14 -19]")#