Do Scalar Product, Dot Product, and Cross Product operands have special names?

Question

Our basic operations have names for their operands:

• Addition: $\rm{Augend}+\rm{Addend}=\rm{Sum}$ -- Generally, we call them both $\rm{Addends}$ or $\rm{Summands}$.
• Subtraction: $\rm{Minuend}-\rm{Subtrahend}=\rm{Difference}$.
• Multiplication: $\rm{Multiplicand}\times\rm{Multiplier}$ -- Generally, we call them both $\rm{Factors}$.
• Division: $\rm{Dividend}\div\rm{Divisor}=\rm{Quotient}$.
  
  

1. But, what about the Scalar Product? We usually write the $\rm{Multiplier}$ on the left, so is the Matrix considered the $\rm{Multiplicand}$, or are there special names? And, would these names apply to the Scalar Product with a Vector?

$$ e\left[ \begin{array}{rr} a & b \\ c & d \end{array} \right] $$

2. What about the Dot Product? And, would these names apply to the Dot Product of Vectors?

$$ \left[ \begin{array}{rr} a & b \\ c & d \end{array} \right] \times \left[ \begin{array}{rr} e & f \\ g & h \end{array} \right] $$

3. Lastly, what about the Cross Product?

Thanks to all.

Answer 1

When mathematicians extend arithmetic to abstract sets like vectors, matrices, and polynomial rings, they continue to use the elementary language. For instance, when we take a matrix $ A $ and write it as $ A = BC $, we call this a factorization of $ A $, where $ B $ and $ C $ are factors whose product is $ A $. Similarly, when we do long division with polynomials, we use the language of dividends, divisors, quotients, and remainders (the latter two are more common, dividend and divisor are not in very common usage in algebra texts).