# 3 vector product

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

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

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i just mean A.B.C

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That doesn't make sense.

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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.

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

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

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

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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)$\neq (A.B).C$

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

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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)$\neq (A.B).C$

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

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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 $\times$.

This doesn't make a lick of sense: $\vec a \cdot \vec b \cdot \vec c$. 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 $\vec a \times (\vec b \times \vec c) = (\vec a \times \vec b)\times \vec c$ is that $\vec c$ is parallel to $\vec a$, i.e., $\vec c = \alpha \vec a$ where $\alpha$ is some scalar. There is no constraint on $\vec b$. If all three are parallel to one another the vector triple product is identically zero for all arrangements of the factors in the product.

oh thanks.

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.

This doesn't make a lick of sense: $\vec a \cdot \vec b \cdot \vec c$. 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 ???

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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.

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This is much easier:

$\vec{a} (\vec{b} \cdot \vec{c})$

Use LaTeX to get the point across.

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$\mathbf{a} \cdot \mathbf{b} \cdot \mathbf{c}$ is not equal to $a(\mathbf{b} \cdot \mathbf{c})$

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 $\mathbf{a} \cdot \mathbf{b} \cdot \mathbf{c}$ this cannot work due to the reason above.

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