You are interpreting AxB for scalars A and B in the sense of the Cartesian product of a pair of sets? [the set of ordered pairs (a,b) where a is an element of A and b is an element of B]. That is rather a stretch since A and B are not sets and their cross product would not be an ordered pair but would, rather, be a [possibly singleton] set of ordered pairs.fresh_42 said:5. Scalar cross scalar to denote the pair (scalar,scalar). "Not valid" is misleading here.
Agreed. I simply found "not valid" a bit harsh, esp. because the word scalar has been put into brackets. Sometimes I've also read lines like ##2 \times 2 = 4##.jbriggs444 said:You are interpreting AxB for scalars A and B in the sense of the Cartesian product of a pair of sets? [the set of ordered pairs (a,b) where a is an element of A and b is an element of B]. That is rather a stretch since A and B are not sets and their cross product would not be an ordered pair but would, rather, be a [possibly singleton] set of ordered pairs.
If one insists on intepreting ##\times## in the sense of the Cartesian product and interpreting 2 in the sense of set theory and the Von Neumann construction then it would be more correct to say that ##|2 \times 2| = 4## and that ##2 \times 2## = { (0,0), (0,1), (1,0), (1,1) }fresh_42 said:Agreed. I simply found "not valid" a bit harsh, esp. because the word scalar has been put into brackets. Sometimes I've also read lines like ##2 \times 2 = 4##.
It works for 7D vectors as well.stevendaryl said:By "x" do you mean the vector cross-product? And by "." do you mean the vector scalar product? If the answer is yes, then those operations are only applicable to a pair of vectors. I would say there are four common types of "multiplication" involving scalars and 3D vectors:
- Scalar times scalar to produce a scalar (ordinary multiplication)
- Scalar times vector to produce a vector (scaling a vector)
- Vector times vector to produce a scalar (scalar or "dot" product)
- Vector times vector to produce a vector ("cross" product)
Zafa Pi said:It works for 7D vectors as well.
A product of scalars refers to the multiplication of two or more scalar quantities, which are single numbers that have magnitude but no direction. In contrast, a product of vectors involves multiplying two or more vector quantities, which have both magnitude and direction.
The product of scalars is a simple multiplication of numbers, while the dot product of two vectors is a mathematical operation that results in a scalar quantity. The dot product takes into account the magnitudes and angle between two vectors, while the product of scalars does not.
No, the product of scalars will always result in a scalar quantity. This is because scalar quantities do not have direction, so they cannot be combined to create a vector quantity.
The product of scalars and vectors is used in various equations and calculations in physics. For example, velocity (a vector quantity) can be calculated by multiplying a scalar quantity (speed) with a vector quantity (direction). It is also used in calculating work, power, and force, among other physical quantities.
No, the product of scalars and vectors is not commutative. This means that the order in which the quantities are multiplied affects the result. For example, the product of a scalar and a vector will give a different result than the product of the vector and the scalar, unless the scalar is equal to 1.