Is the Triple Scalar Product Always Zero?

In summary, the conversation discusses the concept of triple products and their relationship to coplanar vectors. It is clarified that for a triple product to equal zero, all three vectors must be coplanar. It is also noted that if any two coplanar vectors are taken, the cross product will either be zero or normal to both vectors. The textbook specifies that the triple scalar product will equal zero if all three vectors are coplanar, but the accompanying figure may not accurately depict this.
  • #1
member 392791
Hello,

I am confused how vectors that are coplanar will give a triple product of zero? Or is it the case that all 3 vectors must be coplanar for a triple product of zero, or is 2 sufficient? I.e. the vector being dotted with one of the vectors being crossed in the same plane, will this produce a zero scalar product?
 
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  • #2
Take any two of those coplanar vectors. The cross product is either zero or is normal to both of those vectors -- and every other vector that is coplanar with those first two vectors. What's the dot product of two vectors that are normal to one another?
 
  • #3
ok, so I see that the textbook specified that if all 3 vectors are coplanar, then its triple scalar product is zero, which makes sense to me because the projection is going to be zero. It's just that the accompanying figure 3.28 doesn't make me think that the vectors are coplanar.
EDIT: here is the figure that I am referring to
 

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FAQ: Is the Triple Scalar Product Always Zero?

What is the triple scalar product?

The triple scalar product, also known as the scalar triple product or mixed product, is a mathematical operation that takes three vectors and produces a scalar quantity. It is written as (a x b) ⋅ c, and is equal to the magnitude of the cross product of a and b multiplied by the component of c that is perpendicular to the plane created by a and b.

What is the physical significance of the triple scalar product?

The triple scalar product has various physical applications, including calculating the moment of a force about a point, finding the volume of a parallelepiped, and determining the orientation of a plane. It is also used in physics and engineering to solve problems involving vectors and their interactions.

How is the triple scalar product calculated?

The triple scalar product can be calculated using the formula (a x b) ⋅ c = a ⋅ (b x c) = b ⋅ (c x a). To find the volume of a parallelepiped, the magnitude of the triple scalar product is taken, and to calculate the moment of a force, the components of the vectors are used in the formula.

What is the difference between the triple scalar product and the dot product?

The triple scalar product involves three vectors, while the dot product only involves two. Additionally, the triple scalar product produces a scalar quantity, while the dot product produces a vector quantity. The triple scalar product is also used for different applications, such as calculating volume and moment, while the dot product is used for finding the angle between two vectors or projecting one vector onto another.

How is the triple scalar product used in physics?

In physics, the triple scalar product is used to solve problems involving vectors and their interactions. It can be used to determine the torque of a force, the orientation of a plane, and the volume of a parallelepiped. It is also used in mechanics, electromagnetism, and other branches of physics to simplify and solve equations involving three or more vectors.

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