Hello. We have seen in the previous section As matters. When the points are in a plane to obtain the vectors of them. With a number of vectors and total find product. Then we saw two more major transactions. These i, internal e, multiplication. Inner product of vectors and vector vector multiplication. This name comes from the following. Vector by multiplying a vector indeed 're getting. However, a number of inner product obtained from We are. Projection of their inner meaning from multiplying We have the vector obtained by multiplying. We will see their applications. Thus, what we have learned how will be understood to apply both I hope these examples the theoretical structure will consolidate. Sample this:given two vectors. These are spots in space e, he We can look. We want to find the length of this vector. Find the angle between the inner product with we want. By multiplying the angle between the vectors We want to find. Because the inner product of the vector cosine we could find by multiplying the sine of We know. On each of these vectors and we want to find the projection of this the area of the triangle of vectors We want to find. Let's order. Two first and one third of these vectors. 3 as we do. If we go in the direction of x 1 2 2 1 y direction We're going. If we go in the x direction y direction V 1 1 2 3 We're going. These vectors u and v are given to us. They want to find the job size prior to. Easy now. Wherein b is the length of vector lengths b the second component squared plus components of an axis of the frame here because 're getting right triangle with the three teams. Here we see immediately squared plus x2 x1 1 of 4 plus square root of 5. Frames similarly to V The square root of the sum. Now with inner product to find the cosine theta we want. We want to find the interior angle with the cross. The following inner product gives the cosine theta formula. We have found just the length of u. We found V's neck. And there was given vectors. Here we see that E immediately, in the denominator The length of and v is the square root of the length of the square root of 5 10 is there. Denominator of the first component in the first component multiplied with the second component 2 times 1 plus The second component is the product of 1 times 3. Here also is seen here 2 plus 3 of 5 An abbreviation is happening now. If we take into square root is 25, because 5 Or 2 times 50 # 5 of 25 square root of 25, whether he I remove one divided by the square root of 2 is happening. E cosine theta divided by the square root of the square root of 1 2'ys of theta p is at 45 degrees is divided by 4 We find now. Similarly, we also find sinus. Vector multiplied by the size of the individual sine size of the product. We buluyo vector multiplication as follows:First vector are writing. X1, X2. The second vector are writing. Y1, Y2 components. Here, the first vector to the second one. 1 3 second vector. They are the product of E, also from the determinant From the first diagonal 6 minus the first corner of the second diagonal 1. 5 turns. So again, the square root of 5 and the square root of 5 We divide by 10. 45 degrees in here again to provide're getting. Projections. Projections know. There are always three above product. What's in the denominator on the projection yapılıyos to have it. Because I just told you. Projection of v on this. V is independent of the length of this. V is so even if though it bişiy albeit a tiny projection unchanged. Therefore ee, e denominator, there is not is required. E u, u's projection. Here e, here's one projection di root of 5, ba, divided by the square root of 5 As we find 10. U'l and was given situation. 1 2 and 2 3. 2 times 1. 1 times 3. We find it. V. When we're getting the split. If you pay attention to the length of v is v is here v is the length of time that we no longer take part does not stay. Although if it appears in the denominator here v is not the length of stay. V is similarly on When we receive the projection this time involved in different b values. Indeed, we see from this figure. And v is so projected onto the up to 40 percent of its projection on different because the square root of the square root of 5 and 10 There are 2 b of the square root. She also 1.4. If the area of triangle that 2 division of vector multiplication. We found here 5 vector multiplication basis. So it's 2 divided by 5. This is achieved wherein the triangle, these triangles be. Yet more examples of bi. Be perpendicular or parallel. Here the vector b given vector v given. These vectors be orthogonal and for To be parallel alpha We want to find the value. As you can see by one given vector. He now vectors. Here we are going in the x direction 2. 1 2. 1 Y direction we're going. Given the first component of V 1 always. But the latter are not set. So now vek, line, fracture lines all those who might have shown. But the second one below the first component alpha. If the value of alpha according to the above plus below is older As you can see from the two of them they ESI is perpendicular to each other. We want to find it. And b of the two vectors at the same time We want to be parallel to each other. What one first component and the second component must be parallel to the u get? Others may know the rules. If the inner product gives the cosine theta 0 is 0. So theta is pi divided by 2. We see this immediately when you get the inner product the first vector is 2. The second one alpha. By multiplying it times 2 times 1 plus 1 alfa-2a plus alpha equals 0 turns. So the alpha 2 is supposed to be old. So, the vector v is 1 minus 2 we find. We can do this test. Indeed, plus 1 times 2 times 1 minus 2 0 gives. As the vector perpendicular to it It is solid in any other. Therefore, the general solution of a particular is there. In fact, if you look here at these vectors is steep. You still want to extend the vertical will be. Or that you want it be vice versa This will again extend the other. Thus, I have a general solution. These vectors parallel To be in u'yl means that v is parallel to the internal 0 means that multiplication. Because in between will be 0. E sinus angle 0 will be the sine 0 the. Here we are writing the first vector 2 1. The second vector are writing 1 alpha. It is the product of two alpha minus 1. 2 alpha minus one times the vector k is equal to 0. It can be 0 to 1 over the alpha 2 is supposed to be. Put it unless you bring one and one-half 2, we see that. They suck, when we receive the product to hakkaten was the first vector in the second one. This is exactly half the his. Is it 2 by 2 çarpsa is 1. As you can see is a vector in the same direction. General solution can be found here. Is it çarpsa by 2, 2 1 with a katsu is who is possessed. This parallelism using we could find. This is the first way parallelism. P to be parallel from the definition 2 but it is a multiple of the vector parallel to each other happens. In the same direction. The same. Thus, the second one the first floor of one alpha must be. Here we reserve components when 1 2 the alpha will be equal to k will be equal to k. As you can see here, the interests k 1 divided by 2. From the first component. In the second, divided by 2 is 1 alpha understood. So again we find the same vector. Second means to ensure the parallelism method is used. Now here so always work on plane 've done. Why does this have? Easier to understand because his plane the. We çizebiliyo plane. We can better understand what we drew. Our main goal, which is now 3-D go to space. Why? Because all the three-dimensional space We are. Our next session of this generalization will do. You'll see that you have made two-dimensional after The process is quite simple to generalize to three dimensions. But all the events going on in three dimensions. For this reason, a significant generalization. We could start from the third dimension, but the pedagogical the first one in terms of Let me get you there more simple to generalize We chose the easy way.