Triple scalar product question

In summary, the triple scalar product can be used to determine the intersection of normal planes, but it cannot be used to solve anything greater than a 3x3 matrix. It can be used to solve a system of equations or an equation for a variable, but it cannot be used to solve a matrix or a number.
  • #1
DespicableMe
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Homework Statement



Is it possible to use the triple scalar product to solve anything greater than a 3x3 matrix?

Homework Equations


Ax + By + Cz + D = 0


The Attempt at a Solution



In terms of planes, the triple scalar product can be used to determine if the NORMALS of the planes intersect at a point or on a line or in pairs.
But it doesn't necessarily give you the solution, does it?

So is it possible to use the triple scalar product to solve anything greater than a 3x3 matrix?
 
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  • #2
It's not clear what you mean. You can solve a system of equations; you can solve an equation for a variable; you can solve a problem; etc. You don't solve a matrix. That's like saying you solve a number.
 

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 (a single number) as output. It can be used to calculate the volume of a parallelepiped formed by the three vectors, as well as to determine if the three vectors are coplanar.

2. How is the triple scalar product calculated?

The triple scalar product is calculated by taking the dot product of one vector with the cross product of the other two vectors. In other words, it is the dot product of one vector with the normal vector of the plane formed by the other two vectors.

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

The triple scalar product can be interpreted as the volume of a parallelepiped formed by the three vectors. This is because the cross product of two vectors gives a vector that is perpendicular to both, and taking the dot product of this vector with the third vector gives the area of the base of the parallelepiped. Multiplying this area by the height, or the length of the third vector, gives the volume.

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

The triple scalar product has the following properties:

  • It is distributive: a · (b x c) = (a · b) x c = b x (a · c)
  • It is not commutative: a · (b x c) ≠ (a x b) · c
  • It is associative: (a · b) x c = a · (b x c)
  • It is linear: a · (b + c) = a · b + a · c
  • It is invariant under cyclic permutations: a · (b x c) = b · (c x a) = c · (a x b)

5. In what fields is the triple scalar product used?

The triple scalar product has various applications in mathematics, physics, and engineering. It is commonly used in vector calculus, mechanics, and electromagnetism to calculate volume, determine the orientation of planes and angles between vectors, and solve problems involving forces and moments. It is also used in computer graphics and computer vision to calculate the depth of objects in 3D space.

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