Coordinate representation of vectors?

[RJLiberator]

Homework Statement

Starting from the coordinate representation for the vectors, show the result in Equation 1.16 of Griffith's book.

(1.16)$$A \cdot (B \times C) = \left[ \begin{array}{ccc} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{array} \right]$$

Note: Here, I use * to represent dot product and x to represent cross product.

The textbook goes on to state:
$$A \cdot (B \times C) = (A \times B) \cdot C$$
Mod note: Replaced 'x' by \times and '*' by \cdot

Homework Equations

The Attempt at a Solution

I am wondering what the question is actually asking for?

By 'coordinate representation' what do they mean? By looking at vector A as (A_x, A_y, A_z) and same for vector B and vector C. Do I simply work out the math by components and show that the two sides are equal?

 

[Mark44]

Homework Statement

Starting from the coordinate representation for the vectors, show the result in Equation 1.16 of Griffith's book.

(1.16)$$A \cdot (B \times C) = \left[ \begin{array}{ccc} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{array} \right]$$

Note: Here, I use * to represent dot product and x to represent cross product.

The textbook goes on to state:
$$A \cdot (B \times C) = (A \times B) \cdot C$$

Homework Equations

The Attempt at a Solution

I am wondering what the question is actually asking for?

Show that the left side of eqn. 1.16 is equal to the right side.

By 'coordinate representation' what do they mean? By looking at vector A as (A_x, A_y, A_z) and same for vector B and vector C. Do I simply work out the math by components and show that the two sides are equal?

Yes.

 

[RJLiberator]

OK, so the goal is to show $$A \cdot (B \times C) = (A \times B) \cdot C$$ ?

I just worked that out by brute force rather easily.

I guess my confusion was from how the question was stated. "Starting from the coordinate representation for the vectors,...". I also was confused as the book stated (1.16) with this:(1.16)$$A \cdot (B \times C) = \left[ \begin{array}{ccc} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{array} \right]$$

But if the problem was merely to prove
$$A \cdot (B \times C) = (A \times B) \cdot C$$
Then this thread is as good as solved.

 

[Ray Vickson]

OK, so the goal is to show $$A \cdot (B \times C) = (A \times B) \cdot C$$ ?

I just worked that out by brute force rather easily.

I guess my confusion was from how the question was stated. "Starting from the coordinate representation for the vectors,...". I also was confused as the book stated (1.16) with this:(1.16)$$A \cdot (B \times C) = \left[ \begin{array}{ccc} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{array} \right]$$

But if the problem was merely to prove
$$A \cdot (B \times C) = (A \times B) \cdot C$$
Then this thread is as good as solved.

No, you mis-read the question. It first wants you to prove eq. (1.16) --- which, by the way, is a useful result of some importance in itself. Then it asks you to use (1.16) to prove the ABC result stated. The problem was not to "just" prove the ABC result.

 

[Mark44]

One other thing. I believe you have misrepresented the right side of the equation. It should not be a matrix; instead it should be a determinant.

Equation 1.16 should look like this:
$A \cdot (B \times C) = \begin{vmatrix} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{vmatrix}$

 

[RJLiberator]

Ah. Okay guys, thank you for your help. The information I was unaware of was that it was a 3x3 determinant. I know calculated brute force both sides of 1.16 and arrived at the result.