Understanding Triple Scalar Product and Its Properties: Explained Simply

In summary, the dot and cross products can be interchanged in the equation a.(bxc) = b.(cxa) = c.(axb), and this also applies to any three vectors a, b, and c. This property is not explicitly stated in the conversation, but it is necessary to understand why the equation is true.
  • #1
phospho
251
0
Im having trouble understanding this property

my book states that: a.(bxc) = b.(cxa) = c.(axb)

it also states that a.(ax(anything)) = 0

I understand the second point and why that's true, what I don't understand is why a.(bxc) = b.(cxa) = c.(axb) is true

If I name any 3 vectors a b and c would this be true? I'm just really confused to why it works, and my book doesn't really go into depth as it's a high school book, and Wikipedia seems to be vague or too complex for me.
 
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  • #2
phospho said:
Im having trouble understanding this property

my book states that: a.(bxc) = b.(cxa) = c.(axb)

it also states that a.(ax(anything)) = 0

I understand the second point and why that's true, what I don't understand is why a.(bxc) = b.(cxa) = c.(axb) is true

If I name any 3 vectors a b and c would this be true? I'm just really confused to why it works, and my book doesn't really go into depth as it's a high school book, and Wikipedia seems to be vague or too complex for me.

Do you have the property that you can interchange the dot and cross? If not, you should show that first. Then try it on a.(bxc) and see if you can get one of the other forms remembering that the dot product is commutative.
 
  • #3
LCKurtz said:
Do you have the property that you can interchange the dot and cross? If not, you should show that first. Then try it on a.(bxc) and see if you can get one of the other forms remembering that the dot product is commutative.

no, I don't - what is this property?
 
  • #4
LCKurtz said:
Do you have the property that you can interchange the dot and cross? If not, you should show that first. Then try it on a.(bxc) and see if you can get one of the other forms remembering that the dot product is commutative.

phospho said:
no, I don't - what is this property?

a.(bxc) = (axb).c
 

1. What is the triple scalar product?

The triple scalar product, also known as the scalar triple product or box product, is a mathematical operation that takes three vectors as input and produces a scalar (single number) as output. It is represented by the symbol (a x b) · c and is used to determine the signed volume of a parallelepiped formed by the three vectors.

2. How is the triple scalar product calculated?

The triple scalar product can be calculated using the following formula: (a x b) · c = a · (b x c) = b · (c x a) = c · (a x b), where a, b, and c are the three vectors involved in the operation. This formula is also known as the cyclic property of the triple scalar product.

3. What are the properties of the triple scalar product?

The triple scalar product has several important properties, including the following:

  • It is distributive: (a x b) · c = a · c + b · c
  • It is commutative: a · (b x c) = b · (c x a) = c · (a x b)
  • It is anticommutative: a · (b x c) = - (b · (a x c)) = (c · (a x b))
  • It is linear: (ka) · b = k(a · b) = a · (kb)

4. What is the significance of the sign in the triple scalar product?

The sign of the triple scalar product indicates the orientation of the parallelepiped formed by the three vectors. If the result is positive, the vectors are arranged in a counterclockwise orientation, while a negative result indicates a clockwise orientation. A result of zero indicates that the vectors are coplanar.

5. How is the triple scalar product used in physics and engineering?

The triple scalar product has various applications in physics and engineering, including:

  • Calculating the torque on a rigid body
  • Calculating the angular momentum of a particle
  • Determining the orientation of a plane or a line in three-dimensional space
  • Solving problems in mechanics and electromagnetism

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