Vector Calculus: Understanding the Dot Product
The product–process matrix is a model that is used to demonstrate the combination of a product’s volume and various characteristics, together with the nature of the processes that make it. • It is a tool to analyze the relationship between the life cycle of . According to our study book, the Product-Process matrix (PPM) tells us the relationships between different production units and how they are used depending on product volume and the degree of product standardization. This tool assists companies in choosing the best type of production approach based on usage of product, volume being produced, and the amount needed.
Matrix Multiplication Defined page 2 of 3. Just as with adding matrices, the sizes of the matrices matter when we are multiplying. For matrix multiplication to work, the columns of the second matrix have to have the same what does the product- process matrix tell us of entries as do the rows of the first matrix. If, using the above matrices, B had had only two rows, its columns would have been too short to multiply against the rows of A.
Then " AB " would not have existed; the product would have been "undefined". Likewise, if B had had, say, four rows, or alternatively if A had had two or four columns, then AB would not have existed, because A and B would not have been the right sizes.
In other words, for AB to exist that is, for the very process of matrix multiplication to be able to function sensiblyA must have the same number of columns as B has rows; looking at the matrices, the rows of A must be the same length as the columns of B.
You can use this fact to check quickly whether a given multiplication is defined. Write the product in terms of the matrix dimensions. The middle values match:. You can also see this on the dimensions:. Using this, you can see that BA must be a different matrix from ABbecause:. In particular, matrix multiplication is not " commutative "; you cannot switch the order of the factors and expect to end up with the same result.
You should expect to see a "concept" question relating to this fact on your next test. The multiplication works the same way as the previous problem, going across the rows and down the columns. Matrix what is information technology project management is probably the first time that the Commutative Property has ever been an issue. That "rule" probably seemed fairly stupid at the time, because you already knew that order didn't matter in multiplication.
Introducing you to those rules back then was probably kind of pointless, since order didn't matter for anything you were multiplying then. Well, now the Law of Commutativity does matter, because order does matter for matrix multiplication. Always keep in mind that, for matrices, AB almost certainly does what does the product- process matrix tell us equal BA. Stapel, Elizabeth. Accessed [Date] [Month] The "Homework Guidelines". Study Skills Survey. Tutoring from Purplemath Find a local math tutor.
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Jan 22, · The product–process matrix is a model that is used to demonstrate the combination of a product's (or product group's) Volume and Variety characteristics and the nature of the processes that makes likeloveus.com by: 2. Use the product-process matrix to explain that mistake and its likely result. The locomotive manufacturers likely used project technology and processes. This is low volume, high cost production. On the other hand, mass-producing automakers had the technology to make high volume at low per unit cost. What does the product-process matrix tell us? How should the kitchen of a Chinese restaurant be structured? Mar 13 | AM | likeloveus.com Expert's Answer. Previous Next. Related Questions. Recent Questions. More questions Ask a New Question. Attach Files GET ANSWER.
I think of the dot product as directional multiplication. Multiplication goes beyond repeated counting : it's applying the essence of one item to another. For example, complex multiplication is rotation, not repeated counting. And also the justification: "Well Billy, the Law of Cosines you remember that, don't you?
Beyond the computation, what does it mean? The goal is to apply one vector to another. The equation above shows two ways to accomplish this:. The number 3 is "directional growth" in a single dimension the x-axis, let's say , and 4 is "directional growth" in that same direction.
Now, suppose 3 and 4 refer to different dimensions. Let's say 3 means "triple your bananas" x-axis and 4 means "quadruple your oranges" y-axis. Now they're not the same type of number: what happens when apply growth use the dot product in our "bananas, oranges" universe? Applying 0,4 to 3,0 means "Destroy your banana growth, quadruple your orange growth".
But 3, 0 had no orange growth to begin with, so the net result is 0 "Destroy all your fruit, buddy". See how we're "applying" and not simply adding? We're mutating the original vector based on the rules of the second. And the rules of 0, 4 are "Destroy your banana growth, and quadruple your orange growth. Here's how I visualize it:. We list out all four combinations x with x, y with x, x with y, y with y.
The word "projection" is so sterile: I prefer "along the path". How much energy is actually going in our original direction? Take two vectors, a and b. Rotate our coordinates so b is horizontal: it becomes b , 0 , and everything is on this new x-axis.
What's the dot product now? It shouldn't change just because we tilted our head. The common interpretation is "geometric projection", but it's so bland. Here's some analogies that click for me:. One vector are solar rays, the other is where the solar panel is pointing yes, yes, the normal vector.
Larger numbers mean stronger rays or a larger panel. How much energy is absorbed? Photo credit. Take a deep breath, and remember the goal is to embrace the analogy besides, physicists lose track of negative signs all the time. In Mario Kart, there are "boost pads" on the ground that increase your speed Never played? I'm sorry. Photo source. Imagine the red vector is your speed x and y direction , and the blue vector is the orientation of the boost pad x and y direction.
Larger numbers are more power. Neat, eh? Another way to see it: your incoming speed is a , and the max boost is b. The dot product appears all over physics: some field electric, gravitational is pulling on some particle. We'd love to multiply, and we could if everything were lined up. But that's never the case, so we take the dot product to account for potential differences in direction.
It's all a useful generalization: Integrals are "multiplication, taking changes into account" and the dot product is "multiplication, taking direction into account". And what if your direction is changing? Why, take the integral of the dot product , of course! Don't settle for "Dot product is the geometric projection, justified by the law of cosines". Find the analogies that click for you!
Happy math. Learn Right, Not Rote. Home Articles Popular Calculus. Feedback Contact About Newsletter. When dealing with simple growth rates, multiplication scales one rate by another: "3 x 4" can mean "Take your 3x growth and make it 4x larger, to get 12x" When dealing with vectors "directional growth" , there's a few operations we can do: Add vectors: Accumulate the growth contained in several vectors.
Multiply by a constant : Make an existing vector stronger in the same direction. Dot product: Apply the directional growth of one vector to another. The result is how much stronger we've made the original vector positive, negative, or zero. Today we'll build our intuition for how the dot product works. Join k Monthly Readers Enjoy the article? There's plenty more to help you build a lasting, intuitive understanding of math. Join the newsletter for bonus content and the latest updates.
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