Dot and Cross product

davidnobel

🎓 Dot and Cross Product | Vector Algebra Lecture Welcome to this comprehensive video lecture on Dot Product and Cross Product in vector algebra! Whether you're a high school student, university learner, or just brushing up on your math skills, this lesson will guide you through the key concepts, formulas, and applications of these fundamental vector operations.  📌 Topics Covered:  Introduction to Vectors  Understanding the Dot Product (Scalar Product)  Properties and Geometric Interpretation  Introduction to Cross Product (Vector Product)  Right-Hand Rule and Applications  Step-by-step Examples and Practice Problems  🧠 What You’ll Learn:  How to calculate dot and cross products  How to apply these concepts in physics and engineering problems  Real-world examples that highlight their practical use  👨‍🏫 Perfect for students studying:  Physics  Engineering  Mathematics  Computer Graphics and More!  #DotProduct #CrossProduct #VectorAlgebra #MathLecture #Physics #Engineering #Education

Transcript

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

Category

Transcript

Be the first to comment

Recommended