Our inner product of the previous session and vector have achieved the definition of multiplication and thereof geometric projection of the inner product of come equivalent to buying a projection that vector of the product obtained using the two vectors of parallelograms We have seen that the space. Now here's a little bit of this product more features to be compared with each other We'll see. Let's start from the inner product here. Lengths of the inner product of X, Y times We have seen that the cosine of theta. Wherein the components in terms of this first geometric The definition means that the components definition of terms. Multiplying the sum of the components see that. Once these theta angle x of x to y get y in terms of x, y to the following theta. Y from x to the minus theta angle. We know that the negative cosine theta, cosine theta equals. Here we see that already right now per side coefficients E, because it is the product of x y with the cross y'yl inner inner product with X are equal. We see this symmetry. Because here we take into theta times x to y, minus cosine theta was coming. Here x1 y1, x2 y2 would come. As algebraic trigonometric function We observe from the definition of this symmetry. The inner product of X, y. Y'yl equal to the inner product of x. When we look at our own multiplied by y Taking x will be the multiplication of x times x. If you look here cosine theta, theta 0 will be. Therefore, the cosine of theta is 1. Y means that x where x is the same as the multiplied by the square of their height, ie that is obtained. Y1 and Y2 equal to x1 and x2. Therefore, one of the Pythagorean theorem is going to provide. But something important. E is a vector of the length of the inner product the inner product of the vector with itself We find taking. Here, too, is very clear. X the square of the length of X is equal to y is going on. The cosine of theta is falling. So he falls to 1. Because theta E kösinü When 0, 0 a, in radians of the Or the cosine of 0 degrees is 1; theta comes first. E, similarly if you receive theta pi divided by 2, vice versa, where theta 0're getting. I.e. where the pi divided by 2 orthogonal If we take, then the cosine of theta is 0. As you can see when the cosine of theta 0 to X, ie the inner product of y x1 y1 x2 y2 in terms of components plus is equal to 0. This case arises perpendicular vector. Geometric meaning that the projection 've seen. Now let's look at vector multiplication. There vector multiplication sinus. Gene size of the product. In previous sine cosine theta here if Bi theta. Gene components, but the cross product of order. X with the first second of the first with the second There are pros and cons instead of multiplication. Outgoing angle X to Y though the thumb x Open to the E y, than y, then x minus outgoing angle theta is theta. SS, we also know it from trigonometry minus sine theta, theta is negative sine times. To put it once we see that y again, this does not change the inner product of x lengths. Minus sine of theta. It also gives negative. Less times the sine of theta data. She axis, E, X, to us in the y product returns. Therefore, the inner product of X, Y's internal Multiplication of y'l ie the inner product of x being equal to the ordinary Although affected by the negative is happening here. So the anti-symmetric is happening. Öbürkü's going symmetrical. It is very natural. In the first inner product because the cosine There theta. Here are the sine of theta. Gene product itself and ekeke, Consider the product itself. When we receive the product Y'yl x 0 is the angle As you can see to the product of x and x 0 The vector data. Öbürkü has been giving the length of the vector. Because there had cosine theta. Two vectors are parallel, ie, the angle theta 0 still more generally, as it is general y k with him a little more than shock If the sinus because it gives x 0 times theta 0. As you can see exactly the opposite of each other to things, it turns out. That the parallel sided field of 've seen. Now these two bi compile together. A number of X, Y and the projection of the inner product giving. Y in X, E, vector multiplication of a vector giving. See here in multiply, multiplication plain that, in the same O, E multiplication of X, y, gene X, y in multiplied in the same order. We see that the cross came in here. This is a cross coming in determinants can be shown as We can immediately discover. Here the components of the first vector, here the components of the second vector Should we write Determinant of it in the first diagonal product that gives cross-X1 Y2. Here the product in the second diagonal h1 gives x2. Thus, the vector product of the determinants but did you know that in a vector. This vector is perpendicular to the x and y. K is the vector in two dimensions, nu we see. Gene above the previous page As we have seen in X, y the internal y is symmetric but with X vector multiplication anti-symmetric product. Or if y steep inner product X, x is 0 y'yl orthogonal if x their vector product of two vectors is 0 parallel to each other. Have such a complementarity. This is important for the following. Some jobs e, more to do with inner product While suitable for some jobs vector multiplication would be more appropriate for the important. What is the importance of being parallel and perpendicular In terms of applications? Physics, social sciences, probabilities, economic events, such as x and y in the event of a Or an event to force a fiziks will have an effect. Means that x and y are perpendicular to each other There is a projection of x on y. X does not have a projection on the y. So independent events is Shows or that do not affect each other events shows that. X and y are similar to each other in mind parallel to each other on the projection is exactly the same. Because the vector b is parallel to that The projection remains unchanged. Therefore, these vectors in the same direction indicates. That these events, x and In the event of the same nature in which y indicates. As many events will follow a strategy. Comparing XI y y le xA is configured as parallel and perpendicular to x. And so the events of X, which is in the same direction and The events of x do no interaction We may have to divide. As an analogy, it jokingly is made. Caesar, the Roman emperor before we know it senate declared elected president of the Empire The Caesar E, according to the method of divide and conquer conquest his It is said that work to conquer France time In Wales in Welsh said. Gal E, E, unity, e, e, for obama then divide each one by dropping have achieved one by beating. In this simulation in math joke A significant positive things are said. If disassembly events more I can easily understand. For him, this is vertical and parallel here While it may appear as a simple calculator E, that are used in many applications is the approach to be able to understand the events. We saw the beginning of vectors i and j of i and j If we account for the length of the vector first component squared plus wherein the first frame of the second component component squared plus including the square of the second component is seen to be of length 1. When we first take the inner product 0 multiply the first component to the component. The second component with the second component We bumped 0. So the inner product 0. We know already that they are perpendicular. This is going to provide a test. If we consider the vector product of gene i We are writing 1 0. We are writing Jiu 0 1. In this two binary diagonal determinants one of those over multiplication. 0 on Öbürkülerin. Carpinien also k'yl of e, i'yl of jia k is multiplied consistency as we see. I, j, k is orthogonal vectors for they form a triad and other to influence each other, affect. We see this as they steep. And that they are independent from each other In this way we see. Miraculous signs but what we do is all consistent showing calculations. Thereafter, until now always theoretical 've done. We have created an infrastructure. Now we will see examples of them. Internal and vector inner product and vector inner and We'll see about product samples. For now, time to get to know you a little bit of this general theory, theoretical structure To find out, but that this theoretical structure Beispiele to reinforce a better understanding than by will be possible.