Learn about Dot vs. Cross Product
Khan Academy Presents: Understanding the differences between the dot and cross products
Learn about Dot vs. Cross Product
Let’s do a little compare and contrast between the Dot Product and the Cross Product. So, let me just make two vectors, let me visually draw them and maybe if we have time we actually figure out those two things. Well, we actually figure out some Dot and Cross Products will real vectors and let me make another one. I’’ always make a relatively acute angle and maybe I’ll make an, I was do the acute angle. Okay. Let’s called the first one let me label it and take forever if I keep switching colors. Let say angle between them. Okay.
So, let just go over the definitions and then we work on the intuition and hopefully you have little bit on both already. So, what this a.b? Well, first of all that’s the exact same thing as b.a. Order is not the matter when you take the dot product because you end with just a number. And that is equal to the magnitude of ‘a’ times the magnitude of ‘b’ times the cosine of the angle— between them is— fair enough.
Let’s look at the definition of the cross product. What is a cross b? Well, first of all that does not equal to b cross a. It actually equal the opposite direction are give you this a negative of ‘b’ cross ‘a’ because the vector that you end up would ends up flip which ever order you doing in it. But a cross b, that is equal to the magnitude of vector a times the magnitude of vector b. So, fat it looks a lot like the Dot Product but this is where they diverge us as times the sign of the angle between time them. The sign of the angle between them and this is what it really diverge us.
When we talk the Dot Product, we just ended up with a number and this is just a number. There is no direction here. This is just a scalar quantity. But the Cross Product, we take the magnitude of a times the magnitude of b times the side of the angle between them and that provides them activates that also has a direction. And that direction is provided by this normal vector which all of this, it's a unit vector. The unit vectors get a little that on it. It’s a unit vector and what direction this it? Well, that’s defined by the right hand rule.
This is a vector, its perpendicular to both a and b. Then when you might say okay, ‘a’ and ‘b’ the way I draw them. They are both sitting in the plane of this video screen or to your video screen. So, in order for something to be perpendicular to both of this, it either has to pop out of the screen or pop in to the screen, right and when you learn about the Cross Product, it says there is two ways of to solve vector pop out of the screen and you looks like because as a tip of a narrow. And to show a vector going into the screen it's’ like that because that is the back of a narrow, the rear end of a narrow.
So, how do you know which of this two what is because both of this vectors are perpendicular to a and b. Well, that’s where you take your right hand, and then you use a right hand rule. So, you take your index finger in the direction of a, your middle finger in the direction of b and then your thumb points it the direction of n. So, let’s do that. I’m looking at my hand and it looks like that, it takes not an easy thing to do with your right hand. But your hands going to look something like this, your index finger is going to, let me actually do it in the direction of a.
So, your index finger will go in the direction of a, your middle finger goes in the direction of b. So, that is my middle finger and then my other two fingers is just do what they need to do. I like to just spend them out of the way trying my best to draw. So, they just crawl around my hand and then what direction is my thumb? My thumb, well actually I’ll draw it in the wrong angle. My thumb is actually going in this direction, into the page this is the top of my hand. And these are like my veins, right. Or if I actually do it correctly where it you would see your hand from the sides. So, it looks like this.
So would see your pinky and your too like you that your palm and your pinky would be like that. Then you’re other fingers like this. Your middle would go on the direction of b, you index finger goes in the direction of a and you actually wouldn’t even see your thumb because your thumb is pointing straight down. But it think you get the point that a cross b, this n vector is pointing straight down. It’s a unit vector and this provides the magnitude. Unit vector just means it has a magnitude of one. So, the magnitude of the Cross and the Dot Products seem pretty close. They both have the magnitudes of both vectors there.
That product cosine data that Cross Product sine of data but in the huge difference is that sine of data has a direction. It is in different vectors that are perpendicular to both of this. Now let’s get the intuition and if you’ve watch the videos on the Dot and the Cross Product, I hopefully you have a little intuition. But I’ll review it because I think it all fits together when you see them with each other. Let me do some erase, you know, that’s now I wanted to do. Now I am going to erase this way. Okay.
So, first let study with ab cosine of data. So, if you watch the Dot Product video, the cosine of data, if you took let say of b cosine of data. What is b cosine of data? B cosine of data, you can work it all on your own time if you say cosine is adjacent over hypotenuse. B cosine of data is actually go at the magnitude of b cosine is actually going to be the magnitude if you draw up a perpendicular on different color here. If you draw up the perpendicular here, this length right here that’s b cosine of data. Let me draw it separate and I don’t want to mess up this picture too much.
So, if that’s b, now we get the same a and now I was going to use the line tool. If that’s a and now the rest of it, I’ll just do in one color to save time. That’s b, that’s a and this data. B cosine data, if you draw a line perpendicular to a and this is right angle. B cosine of data, adjacent over hypotenuse is equal to cosine data. So, it be the projection of b going in the same direction as a. So, it would be this magnitude. That is b cosine data. The magnitude of that vector right there is b and the magnitude of b cosine of data.
So, when you taking the Dot Product at least the example I just did it. Well, if you view this, the magnitude of a times the magnitude of b cosine data and you’re saying what part of b goes in the same direction as a. And whatever that magnitude is, let me just multiply that times the magnitude of a and I have the Dot Product. Let’s take the piece that they are going to the same direction and multiply them. So, how much do they move together or do they point together or you could view the other way.
You can view the Dot Product as and I think this is the Dot Product video, you can view this a cosine of data b, right. Because it doesn’t matter, these are all scalar quantities. So, doesn’t matter what order you take the multiplication in and a cosine data is the same thing. It's the magnitude of the a vector that’s going in the same direction as b. The projection of a on to the b. So, this vector right here is a cosine data. The magnitude of a cosine data and they are actually the same number if you take how much of vehicle in the direction of a and multiply that with the— of the a. They give you the same number as how much of a goes in the direction of b and they multiply the two magnitudes.
Now what is ab sine data? Well, if this vector right here is a cosine data and you learn this when you learn how to take the components of vectors. And this vector is the magnitude of a sine data. Right, you could rewrite this as magnitude of a, sine data time the magnitude of a in the normal vector direction. So, if you take a sine data times b, you’re saying what pat of a does go in the same direction as b. What part of a completely perpendicular to b, has nothing to do with b. They share nothing in come and then it goes a complete different direction that a sine data and so you take the product of this with b and then yo get a third vector.
It almost says how different are this two vectors and it points in a different direction. It going to give you this sometimes it just called the vector because it applies to some concepts and they are vectors. But the most important of this concept is torque when we talk about the force of a magnetic field and an electric charge. This are all forces or this are all physical phenomenon where what matter isn’t the direction of the force with anther vectors is the direction of the force perpendicular to another vector. And so that’s where the Cross Product comes in usual.
Anyway hopefully that give you a little intuition and you could have good way and you can written this as b sine data and then you would say, oh, that’s the component of b that is perpendicular to a. So, b sine data actually it would have been at this vector, right. Let me draw it here and then it makes more sense. This would be b sine data. So, you could switch or as you could visualize either of the way because they said this is the magnitude of b that is completely perpendicular to a multiply the two and use the right hand rule to get that normal vector.
And that right hand rules that’s nothing and we just decide that were going to use our right hand rule to have a common convention. But people going to use the left hand rule or you might have use it in a different way. It's just the way that we have a consistent, framework that when we take the Cross Product, we all know what direction that normal vector is pointing it.
Anyway in the next video I’ll show you how to actually compute the Dot and Cross when you’re given them in their component notation. See you in the next video
Learn about Dot vs. Cross Product
Today is a good day for all Dota fans. Newest announcement by Valve has made many online gamers happy, Dota 2 is coming in 2011. Dota 2 (Defense of the Ancients) is soon to become, as some say, as great as World of Warcraft and StarCraft....
DotNetNuke is an excellent web development application that makes the web development services absolutely apt for all the purposes....
.NET framework is collection of various technology and development tool. It is very popular cross platform framework. ASP.NET application development is used all around the world for web application development. Rapidsoft Technologies from India provides complete solution for asp.net development at competitive price....
If you\'re on this page you\'re probably wondering if Stella and Dot is Scam or Legal. Well this review is for you. Just like you, thousands of people are wondering the same thing....
Connecting with people is like an infinite connect the dot puzzle that we all played as a child each line you connect gives you a better idea of what the picture looks like and the more lines you connect the more complete that image becomes. The top ten percent of leaders understand the value in not only connecting with their people but consistently reconnecting with them daily in order to maximize the talents they bring to the job....
The head, as home to the brain, is a vital part of the human body. See the different reasons to wear a DOT helmet....
The business world has seen lots of changes over a period of time. With the introduction of new ideas and technologies it has become more and more organized and very lucrative....
In the motorcycling circuit, DOT refers to the Department of Transportation, which regulates helmet circulation by way of ensuring they meet safety requirements. It has made rules to ascertain that all helmets are checked for safety and labels those it considers safe. So if you are looking for a helmet, make sure it has the DOT tag on it, if you don't want to get hauled up for using an illegal helmet....
This article discusses Dot Tel Domain directories. Directories are one of the best ways to make money from the domains. Directories can be created for products, geographical areas, services, and for hybrids of the former categories. The reader will learn about building a Dot Tel Directory in this article....