Understanding the Triple Scalar Product in Vector Calculus

In summary, the formulas A x (B dot C) and (A x B) dot C are both examples of the scalar triple product, which is a combination of a cross-product and a dot-product. It is only defined if the cross-product is done first, and neither (A dot B) x C nor A x (B dot C) is defined if x denotes the vector product.
  • #1
brotivation
12
0

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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  • #2
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
 
  • #3
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)?
 
  • #4
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.
 

1. What is the definition of the triple scalar product?

The triple scalar product, also known as the scalar triple product or mixed triple product, is a mathematical operation that takes three vectors as inputs and produces a scalar (single number) as its output. It is defined as the dot product of one vector with the cross product of the other two vectors.

2. How is the triple scalar product calculated?

To calculate the triple scalar product, first take the cross product of two of the vectors. Then, take the dot product of the resulting vector with the third vector. The resulting number is the triple scalar product. Another way to remember this is "dot product of one vector with cross product of the other two".

3. What is the geometric interpretation of the triple scalar product?

The triple scalar product can be interpreted as the volume of the parallelepiped formed by the three vectors. This means that it represents the amount of space enclosed by the three vectors.

4. What are some applications of the triple scalar product?

The triple scalar product has many applications in physics and engineering. It is used in calculating torque, angular momentum, and moments of inertia in mechanics. It is also used in the calculation of magnetic flux in electromagnetism. In mathematics, it is used in finding the equation of a plane in three-dimensional space.

5. What are some properties of the triple scalar product?

The triple scalar product has several important properties, including the distributive property, the cyclic property, and the anticommutative property. It is also equal to the determinant of a 3x3 matrix formed by the three vectors. Additionally, the triple scalar product is zero if and only if the three vectors are coplanar (lie in the same plane).

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