3 vector product

[swaha]

what is A.B.C? where A,B,C are vectors.

[D H]

Do you mean the scalar triple product [math]\vec a \cdot (\vec b \times \vec c)[/math], or the vector triple product [math]\vec a \times (\vec b \times \vec c)[/math]?

[swaha]

i just mean A.B.C

[D H]

That doesn't make sense.

[swaha]

ctually i saw it in a book, may be a misprint then. shall wait fr a few more negative answers before requesting the moderators to close.

[D H]

First off, swaha, do you understand the difference between the inner product between two vectors [math]\vec a \cdot \vec b[/math] and the scalar product [math]\vec a \times \vec b[/math]?

 

There are two products of three vectors in three-space. I named both in post #2, perhaps a bit to tersely.

 

The first is the scalar triple product [math]\vec a \cdot (\vec b \times \vec c)[/math]. Since the inner product is a commutative operation, this is the same as [math](\vec b \times \vec c)\cdot \vec a[/math]. One could eliminate the parentheses in these forms because [math]\vec a \cdot \vec b \times \vec c[/math] has only one viable interpretation. One geometric interpretation of this product is the volume of a parallelepiped with sides specified by the vectors [math]\vec a[/math], [math]\vec b[/math], and [math]\vec c[/math]. Rearrangements (permutations) of the vectors [math]\vec a[/math], [math]\vec b[/math], and [math]\vec c[/math] might change the sign of the result, but never the absolute value.

 

The second triple product is the vector triple product [math]\vec a \times (\vec b \times \vec c)[/math]. Unlike the scalar triple product, those parentheses are essential here. Specifying things in the right order is also essential. In other words, [math]\vec a \times (\vec b \times \vec c)\ne(\vec a \times \vec b) \times \vec c\ne \vec b \times (\vec a \times \vec c)[/math], and so on. One use of the vector triple product is to compute the component of a vector normal to vector. Suppose [math]\hat a[/math] is a unit vector in [math]\mathbb R^3[/math] and [math]\vec b[/math] is some other vector in [math]\mathbb R^3[/math]. The component of [math]\vec b[/math] normal to [math]\hat a[/math] is [math]\hat a \times (\vec b \times \hat a)[/math].

[triclino]

i just mean A.B.C

 

 

If B and C are on the same currier ,or parallel to A ,then :

 

A.B.C = A.(B.C) =(A.B).C = C.(B.A) =(C.B).A = B.(A.C) ........e.t.c e.t.c

 

Otherwise : A.(B.C)[math]\neq (A.B).C[/math]

[D H]

Use the right nomenclature, please. There are two products defined for vectors in [math]\mathbb R^3[/math], the inner product and the cross product. Neither is denoted with a period.

[swaha]

If B and C are on the same currier ,or parallel to A ,then :

 

A.B.C = A.(B.C) =(A.B).C = C.(B.A) =(C.B).A = B.(A.C) ........e.t.c e.t.c

 

Otherwise : A.(B.C)[math]\neq (A.B).C[/math]

 

why? pls explain. i think its so when they are perpendicular not parallel.

[D H]

Ignore triclino. What he wrote doesn't make sense. Please, people. Learn to use the correct nomenclature. There are two well-defined products for 3-vectors, the scalar product denoted by a center dot, and the cross product denoted by [math]\times[/math].

 

This doesn't make a lick of sense: [math]\vec a \cdot \vec b \cdot \vec c[/math]. That can only mean triclino was talking about the cross product, and what he wrote isn't correct for that either.

 

The correct condition under which [math]\vec a \times (\vec b \times \vec c) = (\vec a \times \vec b)\times \vec c[/math] is that [math]\vec c[/math] is parallel to [math]\vec a[/math], i.e., [math]\vec c = \alpha \vec a[/math] where [math]\alpha[/math] is some scalar. There is no constraint on [math]\vec b[/math]. If all three are parallel to one another the vector triple product is identically zero for all arrangements of the factors in the product.

[swaha]

oh thanks.

[triclino]

.

 

This doesn't make a lick of sense: [math]\vec a \cdot \vec b \cdot \vec c[/math]. That can only mean triclino was talking about the cross product, and what he wrote isn't correct for that either.

 

 

Why you did not ask me what i meant ,but make such a fuss over minor details ??

 

This is a physics forum and people know what a dot product is , and very easily can understand that:

 

A.(B.C) is really A(B.C) since the dot product is always a scalar.

 

Now is not true that if the vectors are on the same currier or parallel then :

 

A(B.C) =(A.B)C ???

[D H]

You left out your "otherwise" in post #7 from the above, triclino.

 

Furthermore, using A.B for the cross product is very, very bad form. That period looks a lot more like a dot than a cross. This is not a minor detail since there are many products for vectors. For example, the inner or dot product, the cross product for vectors in 3- and 7- space, the outer product, the exterior or wedge product, etc. Each has its own symbol and none of them is denoted with a period.

[Cap'n Refsmmat]

This is much easier:

 

[math]\vec{a} (\vec{b} \cdot \vec{c})[/math]

 

Use LaTeX to get the point across.

[darkenlighten]

[math] \mathbf{a} \cdot \mathbf{b} \cdot \mathbf{c} [/math] is not equal to [math] a(\mathbf{b} \cdot \mathbf{c})[/math]

 

Because you cannot dot a vector and a scalar, you can multiply them, but not dot them. This is kinda trivial, but the reason I say it is if there was maybe some type of proof or equation that had a similar form, you would not be able to an operation like this.

 

Or whatever you are trying to say but either way [math] \mathbf{a} \cdot \mathbf{b} \cdot \mathbf{c} [/math] this cannot work due to the reason above.