Question

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

Answer 1

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>`