Question

i got this question: Prove
a cross (b cross c)= (a dot c)b-(a dot b)c

Answer 1

what i tried doing was working the LHS, i first did b cross c, the i did a cross (bxc), i dont know if that how i should proceed

Answer 2

then for the RHS, i did first a dot c, but i am comfused as to whether the b on the ouside of the parenthese is distrubued into the dot product or not.

Answer 3

a.c is a scalar; so is a.b

Answer 4

\[(<a_1,a_2,a_3,...,a_n>.<b_1,b_2,b_3,...,b_n>)<c_1,c_2,c_3,...,c_n>\]
\[(a_1b_1+a_2b_2+a_3b_3+...+a_nb_n)<c_1,c_2,c_3,...,c_n>\]
\[<a_1b_1c_1+a_2b_2c_1+a_3b_3c_1+...+a_nb_nc_1,a_1b_1c_2+a_2b_2c_2+a_3b_3c_2+...+a_nb_nc_2,\]
\[\hspace{15em} ...,a_1b_1c_n+a_2b_2c_n+a_3b_3c_n+...+a_nb_nc_n>\]

Answer 5

i received a text from a friend who is taking the class wth me, and its an extremely long proof

Answer 6

it can be, yes

Answer 7

what do you mean "it can be"

Answer 8

as opposed to "it cant be" i would say that its more probable than not.

Answer 9

i was looking at some sources on the web, and it talked about the possiblility that we could use symmtery to prove this, by working with the x-axis or somehting like that

Answer 10

\[(a.c)b\]
\[<a_1c_1b_1+a_2c_2b_1 +a_3c_3b_1+...+a_nc_nb_1, a_1c_1b_2 +a_2c_2b_2 +a_3c_3b_2+...+a_nc_nb_2,\]
\[\hspace{15em} ...,a_1c_1b_n+a_2c_2b_n+a_3c_3b_n+...+a_nc_nb_n>\]

\[(a.b)c-(a.c)b\]

\[\binom{(a_1b_1c_1+a_2b_2c_1+a_3b_3c_1+...+a_nb_nc_1)}{-(a_1c_1b_1+a_2c_2b_1 +a_3c_3b_1+...+a_nc_nb_1)},\binom{(a_1b_1c_2+a_2b_2c_2+a_3b_3c_2+...+a_nb_nc_2)}{-(a_1c_1b_2 +a_2c_2b_2 +a_3c_3b_2+...+a_nc_nb_2)},\]
\[\hspace{15em} ...,\binom{(a_1b_1c_n+a_2b_2c_n+a_3b_3c_n+...+a_nb_nc_n)}{-(a_1c_1b_n+a_2c_2b_n+a_3c_3b_n+...+a_nc_nb_n)}\]

Answer 11

if you can get the left to look like that

Answer 12

I think you are misinterpreting. The cross product, a specific form of an outer product, is only for vectors in R^3. This means that if you were to assume a general form you wouldn't assume
\[\vec{A}=(a_1,a_2,...a_n)\]
You would assume:
\[\vec{\Omega}=(\Omega_1,\Omega_2, \Omega_3)\]

This should help.
Also, wikipeidia quote:
"In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and therefore normal to the plane containing them. The name "cross product" is derived from the cross symbol "×" that is often used to designate this operation; the alternative name "vector product" emphasizes the vector (rather than scalar) nature of the result. It has many applications in mathematics, engineering and physics."