Understanding the Triple Scalar Product in Vector Calculus

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Homework Help Overview

The discussion revolves around understanding the triple scalar product in vector calculus, specifically the expressions A x (B dot C) and (A x B) dot C. Participants are exploring the relationships and definitions of vector operations, particularly focusing on the distinction between dot and cross products.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are questioning whether A x (B dot C) can be interpreted as a valid operation, given that B dot C results in a scalar. There is discussion about the definitions of the dot and cross products and their applicability in the context of the formulas presented.

Discussion Status

Some participants have provided clarifications regarding the definitions of vector operations, noting that the first expression is not a valid cross product. There is an ongoing exploration of how to correctly interpret the operations involving scalars and vectors.

Contextual Notes

Participants are navigating the constraints of vector operations and the definitions that govern them, particularly in relation to homework guidelines that may limit the types of operations that can be performed.

brotivation
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Homework Statement



A x (B dot C)

(A x B) dot C

They are vectors.

Homework Equations



A x (B dot C)

(A x B) dot C

The Attempt at a Solution



I know how to do my homework, but I am confused on these formulas.

Is the first formula "A x (B dot C)" the same as the second one? I know the second one is the same as
A dot (B x C).

It doesn't make sense to me. Wouldn't the B dot C become a scalar? So how could A cross with that?
 
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brotivation said:
A x (B dot C)

The first multiplication is not "cross product". The dot product of two vectors is a scalar, and the cross product is defined for two vectors. What you wrote is just a product of A with a scalar.

ehild
 
ehild said:
The first multiplication is not "cross product". The dot product of two vectors is a scalar, and the cross product is defined for two vectors. What you wrote is just a product of A with a scalar.

ehild

So would that mean I do A multiply by (B dot C)?
 
brotivation said:
So would that mean I do A multiply by (B dot C)?
The scalar triple product of for vectors A, B, and C is a combination of a cross-product (also called a vector-product) and a dot-product (also called a scalar-product) .

It's only defined if you do the cross product first.

(A×B)∙C and A∙(B×C) are both defined.

Neither (ABC nor A×(BC) is defined if × denotes the vector product. It's not possible to form a cross-product with a vector and a scalar.
 

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