Coordinate representation of vectors?

In summary: I also calculated the determinant of the right side of 1.16 to arrive at the ABC result.In summary, the goal of this problem was to prove equation 1.16 from Griffith's book, which states that ##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}##, and to use this result to prove that ##A \cdot (B \times C) = (A \times B) \cdot C##. It was necessary to understand that the coordinate representation for vectors refers to expressing them in terms of their components, and
  • #1
RJLiberator
Gold Member
1,095
63

Homework Statement


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

(1.16)[tex]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][/tex]

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

The textbook goes on to state:
[tex] A \cdot (B \times C) = (A \times B) \cdot C [/tex]
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?
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
RJLiberator said:

Homework Statement


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

(1.16)[tex]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][/tex]

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

The textbook goes on to state:
[tex] A \cdot (B \times C) = (A \times B) \cdot C [/tex]

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.
RJLiberator said:
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.
 
  • Like
Likes RJLiberator
  • #3
OK, so the goal is to show [tex] A \cdot (B \times C) = (A \times B) \cdot C [/tex] ?

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)[tex]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][/tex]

But if the problem was merely to prove
[tex] A \cdot (B \times C) = (A \times B) \cdot C [/tex]
Then this thread is as good as solved.
 
  • #4
RJLiberator said:
OK, so the goal is to show [tex] A \cdot (B \times C) = (A \times B) \cdot C [/tex] ?

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)[tex]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][/tex]

But if the problem was merely to prove
[tex] A \cdot (B \times C) = (A \times B) \cdot C [/tex]
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.
 
  • Like
Likes RJLiberator
  • #5
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}##
 
  • Like
Likes RJLiberator
  • #6
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.
 

FAQ: Coordinate representation of vectors?

1. What is a coordinate representation of a vector?

A coordinate representation of a vector is a way of expressing the magnitude and direction of a vector in terms of its components along a set of axes. This representation is typically shown as an ordered pair or triple of numbers, where each number represents the length of the vector in a specific direction.

2. How do you determine the coordinates of a vector?

The coordinates of a vector can be determined by using the Pythagorean theorem and basic trigonometry. First, the length of the vector is found by taking the square root of the sum of the squares of its components. Then, the angle between the vector and a chosen axis is calculated using trigonometric functions. Finally, the coordinates are calculated using the length and angle of the vector.

3. What is the significance of the coordinate representation of vectors?

The coordinate representation of vectors is significant because it allows for the easy visualization and manipulation of vectors in a two or three-dimensional space. It also allows for the use of vector operations, such as addition and subtraction, to be performed algebraically.

4. How does the orientation of axes affect the coordinate representation of vectors?

The orientation of axes affects the coordinate representation of vectors by changing the direction and magnitude of the vector's components. For example, if the axes are rotated, the coordinates of the vector will change, even though the vector itself remains the same.

5. Can the coordinate representation of vectors be extended to higher dimensions?

Yes, the coordinate representation of vectors can be extended to any number of dimensions. In higher dimensions, vectors are represented as ordered sets of numbers, with each number representing the component of the vector in a specific direction. The Pythagorean theorem and trigonometry can still be used to determine the coordinates of a vector in higher dimensions.

Similar threads

Replies
9
Views
2K
Replies
11
Views
2K
Replies
6
Views
2K
Replies
4
Views
1K
Replies
5
Views
1K
Replies
17
Views
3K
Back
Top