What is the cross product of #<-3 ,-6 ,-3 ># and #<0 ,1 , -7 >#?

1 Answer
Miguel F.
Apr 27, 2016

We will use determinants to calculate cross product.

Explanation:

First of all, let us rewrite both vectors, in terms of the vectors of the basis of #mathbb(R)^3#: #{vec{e_1}, vec{e_2}, vec{e_3}}# (or you may use #{vec{i}, vec{j}, vec{k}}#).

  • #<-3, -6, -3> = -3 vec{e_1} -6 vec{e_2} - 3 vec{e_3}#
  • #<0,1,-7> = vec{e_2} - 7 vec{e_3}#

Now, cross product of two vectors #< x,y,z ># and #< x', y', z' ># is given by:

# < x, y, z > times < x', y', z'> = det ((vec{e_1}, vec{e_2}, vec{e_3}), (x, y, z), (x', y', z')) #

In our case:

# < -3, -6, -3 > times < 0, 1, -7 > = #

#= det ((vec{e_1}, vec{e_2}, vec{e_3}), (-3, -6, -3), (0, 1, -7)) =#

#= vec{e_1} cdot [(-6) cdot (-7) - (-3) cdot 1] -#
#- vec{e_2} cdot [(-3) cdot (-7) - (-3) cdot 0] +#
#+ vec{e_3} cdot [(-3) cdot 1 - (-6) cdot 0] =#

#= 45 vec{e_1} - 21 vec{e_2} - 3 vec{e_3} =#

#= < 45, -21, -3>#

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