Why is used in dot product

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. Round the answer to two decimal places. We get. The ship is moving at Repeat the previous example, but assume the ocean current is moving southeast instead of northeast, as shown in the following figure.

Now that we understand dot products, we can see how to apply them to real-life situations. The most common application of the dot product of two vectors is in the calculation of work. From physics, we know that work is done when an object is moved by a force. We saw several examples of this type in earlier chapters. Now imagine the direction of the force is different from the direction of motion, as with the example of a child pulling a wagon.

To find the work done, we need to multiply the component of the force that acts in the direction of the motion by the magnitude of the displacement. The dot product allows us to do just that. The customary unit of measure for work, then, is the foot-pound. One foot-pound is the amount of work required to move an object weighing 1 lb a distance of 1 ft straight up.

Find the work done by the conveyor belt. The distance is measured in meters and the force is measured in newtons. What is the work done by this force? 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.

We use the dot product of course! Finally, we conclude that the dot product plays a key role in the transformation of a vector from one basis to another and that the dot product is hidden in the definition of matrix multiplication in that one view of a matrix-vector product is that each element in the product represents a dot product between a row of the left and a column of the right.

The dot product is an essential ingredient in matrix product. The geometric idea of the dot product has been touched upon, but there is a vast generalization of this product in geometric algebra , the algebra of not only oriented lines vectors but planes, volumes, and more called blades. We throw away the normal part, and the previous logic applies for the tangential part. In this light, the dot product of vectors may actually be the most non -intuitive part of this reasoning.

When one calculates A. B , two measurements happen: measurement of how small the angle between them is, and how long A and B are. B basically means projection length of A on B , with this length then scaled by the absolute length of B.

One way to think about the interpretation of the dot product is to think how would one maximise or minimise the dot product between two vectors. Let's assume we are trying to maximise the dot product between two vectors that we can modify:. The dot product will be grow larger as the angle between two vector decreases. The dot product A. B will also grow larger as the absolute lengths of A and B increase.

This is because as A gets larger, its projected length will be longer, and as B 's length gets larger, the scaling of A 's projection will grow larger, given that B 's absolute length will act as a scaler of A 's projection length. Hence in problems where is desirable to maximise or minimise the size of vectors and minimise the deviation or angle between them, quantifying it using the dot product can be useful.

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Create a free Team What is Teams? Learn more. What is the use of the Dot Product of two vectors? Ask Question. Asked 8 years, 5 months ago. Active 8 months ago. Viewed 52k times. MJD Antonio Antonio 2 2 gold badges 5 5 silver badges 10 10 bronze badges. This is a better approach than using the cross product as the cross product can only be defined in a few dimensions normally only 3 dimensions.