What Are the Key Differences Between Simple Product and Dot Product?

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The discussion clarifies the differences between simple product and dot product, particularly in the context of complex numbers and vectors. The simple product is not applicable to vectors, while the dot product provides a specific way to multiply two vectors, resulting in a scalar. The formula for the dot product of two three-component vectors is given as the sum of the products of their corresponding components. Additionally, the cross product is mentioned as another vector multiplication method, which results in a vector rather than a scalar. Understanding these distinctions is crucial for proper application in mathematical contexts.
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whats the difference between simple product and dot product?
 
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Products are definitions, they are way that it makes sense to take 'x' of 'y'. With complex numbers this is what I am assuming you mean by "simple product", however the same definition doesn't make sense with vectors. The dot product is just one way of defining a product between two vectors, if we assume they are composed of 3 components the dot product is defined as...
< a_1, a_2, a_3 > \cdot < b_1, b_2, b_3 > = a_1b_1 + a_2b_2 + a_3b_3
An alternative way of defining vector products is the "cross product"
 
I will point out (may it be known) that the cross product of two vectors is a vector where as the dot product of two vectors is a scalar. Two completely different things.
 
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