00:00So, we are going to learn about vector.
00:16The first question is what is vector?
00:20So, vector is a mathematical object that has magnitude and direction and direction.
00:39For example, like let's say the velocity of an object is 10 meter per second in south direction.
00:57So with this example, you can see that this is the magnitude 10 meter per second and direction
01:04is in south direction.
01:06So if it has a magnitude and direction, then this is a vector.
01:12And if it has only a magnitude, then it is a scalar.
01:18So now talking about the what is a vector, the basis vector or before that, let's talk
01:29about the vector space.
01:37Vector space.
01:40So vector space is a set where vector can be added together and multiplied by scalar and
01:47the result is still in the set.
01:51So, like let's say for example, in 3D space, let's first try the definition.
02:04Vector space is a set where vectors can be added together.
02:29And multiplied by scalar and the result is still in set.
02:55So for example, in 3D space, we can add two vectors like 2, 4, 5 plus 2, 4, 5 and the value
03:16will be 2, 2, 4, 4, 4, 4, 8, 5, 5, 10 and also we can like scale a vector like we can multiply
03:29a vector.
03:30So let's say 2 into 1, 2, 4.
03:34So it will be 2, 4, 6.
03:38So this is the vector space.
03:39A vector space is a set where vectors can be added together and multiplied by a scalar and
03:46the result is still in set.
03:48So the first part can be added together.
03:52So here we have added and also like can be multiplied by scalar.
03:55So we have multiplied by scalar.
03:57So this is the vector space.
04:01Now talking about the basis vectors.
04:15To describe vector in n dimension, we need n basis vectors up to describe vectors in n dimensional
04:37space.
04:38So what does this mean that like basic vector or minimal state of vector that can represent
04:57any vector in a space while in a combination.
05:01So let's say if we want to like draw a vector in cody space, then we only need i and j.
05:12And if we want in 3D space, then the basis vector will be i, j, k.
05:21Then we can draw any vector and vector can be written as let's say v is equal to v, x, i plus
05:29v, y, j, plus v, j, k.
05:38So they are linearly independent.
05:42This means that none of the basis vector can be written in terms of further and they are
05:46orthogonal like basis vector are in perpendicular to each other orthogonal.
05:50And each has a length one unit.
05:53So this i, j, k all are perpendicular to each other.
05:58For example, in Cartesian basis, this will be i is equal to 1, 0, 0.
06:08So k vector will be 0, 1, 0 and k vector will be 0, 0, 1.
06:18So now talking about like why do we need a basis vector.
06:25We need a basis vector because any vector in the space can be expressed in terms of them.
06:33And this is much easier than describing vector with arbitrary direction.
06:37So let's say for example, we can define a vector by 2, i, j, plus 3, k, 2, 1, 3.
06:52So this is a much easier than describing vector with arbitrary direction.
06:59So the key idea from this is that vector lives in vector space and they are described using
07:11basis vectors, usually orthogonal in Cartesian coordinate.
07:16And any vector is just a linear combination of basis vector.
07:34So it has three properties.
07:38So first is that they are linearly independent.
07:50Second is that spanning, which means that I will discuss more about that later.
08:01And third is that they are orthogonal or the normality.
08:11So talking about the first linear independent, so they are linearly independent means no vector
08:18in set can be written as linear combination of the other.
08:22So for example, i will be 1, 0 and j will be 0, 1, they are independent.
08:38And so yeah, linear independent means i will be separate and j will be separate and they
08:46can't be written in the form of each other.
08:53So now talking about the spanning, so any vector in the space can be written as a linear combination
08:58of basis vector, any vector in the space can be written as a linear combination of basis vector.
09:28So for example, in 3D, y dash will be y i plus y 2 j plus y 3 k.
09:48And third property is that they are orthonormal to each other.
09:52Like orthonormal means they are 90 to each other, means perpendicular, orthogonal means perpendicular
09:59and normalize this unit vector.
10:02So i dot j will be equal to j dot k will be equal to k dot i and this will be equal to 0.
10:12So dot product, it will cos theta and cos 90 will be 0, hence the value will be 0.
10:21So now talking about why it is called basis, why basis, why basis, why basis, because they
10:32are the foundation to build any vector in space.
10:40Because it is the foundation to build any vector in the space.
11:05So this is that they will be needed to define any vectors that we are going to write.
11:15So in real life example, if we want to write vector in 3D space, then we will need 3 basis
11:21vectors which will be i, j, k.
11:26So final takeaway from this slide is that a vector space needs basis vectors.
11:32They must be linearly independent and idly orthonormal and any vector cartesian basis.
11:47So in 3D cartesian coordinates, we need 3 orthonormal unit vectors which are i, j and k.
12:01So talking about i, so it is 1, 0, 0, j is 0, 1, 0 and k is 0, 0, 1.
12:11So this is the unit vector in x direction.
12:17This is in y direction and this is in z direction.
12:23And they are perpendicular to each other and they are of unit length.
12:30And now like using this i, j, k, we can represent any vector in the space 3D space using this
12:37vector.
12:39And in coordinate form, this will be let's say is equal to v1, v2 and v3.
12:49For example, let's say v is equal to 2y, 1j and 3k.
12:58So the component of x here is 2, component of y here is 1 and component of z here is 3.
13:08So now talking about why cartesian basis is useful.
13:11So it is the simplest and most intuitive basis for real-world engineering problem.
13:17And orthonormal properties make calculation dot and cross product much easier.
13:26So now final takeaway from this is that cartesian basis, i, j, k, forms the building blocks of
13:32the 3D vector.
13:33And any 3D vector can be written as a combination of them.
13:46So the dot product, it is also known as the scalar product between two vectors.
13:52So let's say u dot v.
13:57So the dot product of this will be u, v cos theta.
14:06So magnitude of u, magnitude of v multiplied by cos theta.
14:10So let's say this is theta, this is u, this is v.
14:17So the dot product is like the projection of v on u.
14:22So this is the dot product.
14:28So the input required for this is the two vectors.
14:32And the output we get is a one scalar means a number.
14:37So this is the reason it is called scalar product.
14:41Now talking about the dot product in Cartesian coordinate.
15:01So let's say if u vector is u1, u2, u3 and v vector is v1, v2 and v3.
15:20Then the dot product of u1 dot v1 will be is equal to u1 v1 plus u2 v2 plus u3 v3.
15:38Because the product with other will be cost 90 degree which will be 0.
15:44Hence we will directly calculate that.
15:55And now talking about if we how we will find what will be the dot product of u dot u.
16:04u dot u.
16:06So it will be u1 square plus u2 square plus u3 square will just replace v with u.
16:20So the final takeaway from this is that dot product is the value of dot product is scalar.
16:27And it measures the alignment between the two vectors and yes.
16:49So first talking about algebraic properties.
17:04So the first is commutative which means that u dot v is equal to v dot u.
17:19So where we will be the same.
17:22Second is that distributive which means that u vector multiplied by alpha v1 alpha vector
17:34plus beta w vector will be alpha u dot v plus beta u dot w.
17:45So this is the distributive property.
17:46And now talking about the third which is associated with scalar.
18:01So alpha v dot u vector is equal to alpha u dot v vector.
18:10So the value will be the same.
18:12Now talking about the orthogonality.
18:16So if two vectors are orthogonal to each other then the value will be 0.
18:24If there is 90 degree then the value will be 0.
18:27Since u dot v is equal to u dot u v cos 90 degree.
18:35So this value will be 0.
18:36So if they are orthogonal to each other then the value will be 0.
18:42And now talking about the dot product of basis vector.
18:47So i dot i is equal to j dot j is equal to k dot k.
18:53The value of this will be 1.
18:55And because the degree between them is 0 and i dot j is equal to j dot k is equal to i dot
19:03k is equal to 0 since they are orthogonal to each other.
19:13Now talking about next projection using dot rug.
19:19So let's say this is v vector and this is n vector which is unit vector.
19:39So.
19:52So while calculating the projection.
19:54So the projection value this is theta.
19:57So v dot n will be v and cos theta and like this will be the projection.
20:03So let's take example to better understand this.
20:07So let's say v is equal to 2, 1, 3.
20:12And we want to find the projection of this vector on x axis.
20:18So we will just multiply this vector by i.
20:22So this will be 2, 1, 3 dot 1, 0, 0.
20:28So this will be 2.
20:30So the projection means the length of the x axis is 2 units.
20:36So this is the projection which we are talking about.
20:41Now talking about the next concept which is the flux is the physical example.
20:50So if v is the velocity field and n is the surface normal then v dot n represents the flux across
21:18the surface.
21:29There is even no flux across the surface.
21:33So the final takeaway from this is that the dot product is cumulative distributed and scalar
21:40associative.
21:41Those these are the property.
21:43And if the angle between them is 90 degree.
21:47So the dot product will be 0.
21:50And what geometrically this means that it is the projection on other vector.
22:04I mean if we are doing the dot product this means that we are finding the projection of one
22:10of one basis vector on another.
22:17Cross products cross products.
22:28So first talking about the definition.
22:34So this cross products is also called vector product of vector product.
22:41And this is the two vector u and v is defined as u and v vector is defined as u v sin theta.
22:55Where theta is the angle between them.
22:59So let's say this is u and this is v and this is theta.
23:12So yes and to find the direction like we curl our hand right hand from u toward v and the direction
23:31of the thumb point out the direction of u dot v.
23:36And talking about output.
23:42So input is two vectors is two vectors and output is one vector.
23:53Output is also a vector.
23:56So like the dot product which is a scalar this is a here the output is a vector.
24:03And in dot product it was a scalar.
24:07So now talking about the cross product in Cartesian coordinate.
24:22So let's say u is u 1 u 2 u 3 and v is v 1 v 2 v 3 then the cross product will be written
24:38as u 1 v 2 v 3 and so while expanding this u cross v will get u 2 v 3 minus u 3 v 2 so we'll
25:07multiply by this minus this then minus this then minus this is i minus j it will be u 1 v 3 minus
25:25u 3 v 1 j cap and for k it will be plus k u 1 v 2 minus u 2 v 1 k cap.
25:42So talking about the geometric meaning of this what is the geometric meaning.
25:56So the magnitude of the cross product equals the area of the parallelogram.
26:02So u vector cross v vector is equal to u v sin theta.
26:13So this is the area of the parallelogram and the physical application are to find torque angular momentum
26:31and magnetic force.
26:40So
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