The cosines for these angles are called the direction cosines. So far, we have focused mainly on vectors related to force, movement, and position in three-dimensional physical space. However, vectors are often used in more abstract ways. For example, suppose a fruit vendor sells apples, bananas, and oranges. On a given day, he sells 30 apples, 12 bananas, and 18 oranges. In this example, although we could still graph these vectors, we do not interpret them as literal representations of position in the physical world.
We are simply using vectors to keep track of particular pieces of information about apples, bananas, and oranges. This idea might seem a little strange, but if we simply regard vectors as a way to order and store data, we find they can be quite a powerful tool.
We then add all these values together. So, in this example, the dot product tells us how much money the fruit vendor had in sales on that particular day. When we use vectors in this more general way, there is no reason to limit the number of components to three. What if the fruit vendor decides to start selling grapefruit? In that case, he would want to use four-dimensional quantity and price vectors to represent the number of apples, bananas, oranges, and grapefruit sold, and their unit prices.
As you might expect, to calculate the dot product of four-dimensional vectors, we simply add the products of the components as before, but the sum has four terms instead of three. AAA Party Supply Store sells invitations, party favors, decorations, and food service items such as paper plates and napkins.
During the month of May, AAA Party Supply Store sells invitations, party favors, decorations, and food service items. Use vectors and dot products to calculate how much money AAA made in sales during the month of May. How much did the store make in profit? We have. To calculate the profit, we must first calculate how much AAA paid for the items sold.
Their profit, then, is given by. All their other costs and prices remain the same. If AAA sells invitations, party favors, decorations, and food service items in the month of June, use vectors and dot products to calculate their total sales and profit for June. As we have seen, addition combines two vectors to create a resultant vector.
But what if we are given a vector and we need to find its component parts? We use vector projections to perform the opposite process; they can break down a vector into its components. The magnitude of a vector projection is a scalar projection. We return to this example and learn how to solve it after we see how to calculate projections.
To find the two-dimensional projection, simply adapt the formula to the two-dimensional case:. Sometimes it is useful to decompose vectors—that is, to break a vector apart into a sum. This process is called the resolution of a vector into components. Projections allow us to identify two orthogonal vectors having a desired sum.
Then, we have. Its engine generates a speed of 20 knots along that path see the following figure. In addition, the ocean current moves the ship northeast at a speed of 2 knots.
I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent. The dot product tells you what amount of one vector goes in the direction of another. For instance, if you pulled a box 10 meters at an inclined angle, there is a horizontal component and a vertical component to your force vector.
So the dot product in this case would give you the amount of force going in the direction of the displacement, or in the direction that the box moved. This is important because work is defined to be force multiplied by displacement, but the force here is defined to be the force in the direction of the displacement. By definition:. So, we need to find a link between this and the cosine.
Thus, we only need to show. We can start by using the Pythagorean theorem:. It might help to think of multiplication of real numbers in a more geometric fashion.
For dot product, in addition to this stretching idea, you need another geometric idea, namely projection. I'm being a little careless about plus and minus signs, but those can be incorporated into this picture too. I think of dot product as the "same-ness" of two vectors. If you divide their dot product by the product of their magnitude, that is the argument for an arccosine function to find the angle between them.
My application for the dot product is finding the angle between two vectors for calculating the force required to pull a cable through two or more pipes with a bend. It's hard to do this in a three dimensional world without knowing how to calculate the dot product. Math makes life really easy :. The dot product of two vectors u,v is the area of the parallelogram u,v' where v' is v rotated by 90 degrees.
Dot product is the product of magnitudes of 2 vectors with the Cosine of the angle between them. You can take the smaller or the larger angle between the vectors. That is if theta is the angle then you can take theta as well. In Physics, as an example, Mechanical Work is a scalar and a result of dot product of force and displacement vectors.
Like-wise, Magnetic flux is the dot product of magnetic field and vector area. Let me try to explain this with an example. Say you wish to find the work done by a force F along X axis over a distance d. However the problem also tells you that the direction of the force is not along the X axis but at an angle of 60 degree with X axis. Now you know that the work done is the product of force and displacement.
But in this case you know that the force is not exactly totally acting in the direction of X axis, since it is inclined at 60 degree. So what you can do is, find what is the contribution of this force in the X direction. Well it turns out with simple trigonometry that it is F Cos60 in direction of X axis. This can also be represented as F.
So you see, dot product gives us the magnitude of a certain entity in this case work by way of attributing a certain vector in this case force F in the direction of the other vector. Here "d" was the other vector along which work was being found. What is Dot Product of Vectors.
I don't think the dot product has a very obviously interesting visual interpretation. I'm not a math teacher, but if I were asked to define the dot product in a course, I'd start by defining the scalar projection first. 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.
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