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