"Factors are an attribute of a vector that specifies the possible values and their order."
"Vectors with that property have a special name. They're called eigenvectors."
"There's no harm in trying. As Mr. Aaron Draplin says, vectors are free. Try things, no problem."
"Normalized vectors are very, very special; they're very useful."
"Distance is from one vector to another, while length is the length of a single vector."
"If both of the vectors are normalized, the dot product is guaranteed one and negative one when they're either pointing in the same direction or opposite."
"If you have a vector in any direction that has a length of one, if you multiply that by say three, then you're going to get a vector in the same direction that is three times as long."
"Once you start using vectors, this process isn't as simple anymore."
"And here it's just a vector projected on another vector projected on another vector-- something everybody knows how to do."
"Vectors are used literally everywhere in games."
"If you have two normalized vectors, then doing the dot product between those two can sort of tell you how close they are to being similar versus looking directly away from each other."
"Vector addition if you have two vectors and you want to add these together, it's kind of like taking the moving one or the other to the tip or the other one."
"Vectors don't have the spaces. Vectors are just numbers."
"You can use matrices to transform vectors between coordinate spaces."
"If W is one, you're transforming a point."
"If W is zero, you're transforming only the direction."
"Vectors are incredibly fundamental because it's what we use for almost everything."
"So any vector can be written as a multiple of another vector. That tells me that the two vectors are parallel."
"Vectors can also be added by adding corresponding components."
"The great idea is that if you have any point on the plane and you draw the vector from your favorite point... to any other point on that plane, that vector is in the plane and so it's perpendicular to their normal vector."
"Vectors are not limited to particular positions, so you can go from A to B, and you can go from C to D, and it's the same length and the same direction."
"You can do maths with vectors, adding, subtracting, and even using multiples."
"This is a nice alignment with my Nature of Code materials where we look at, I look at through those tutorials 2D vectors and how to do vector math like add subtract normalize distance all that stuff."
"...so there's a relation between basis vectors for m two by three and basis vectors for r two cross r three."
"If one vector can be expressed as a constant multiple of another one then they must be parallel."
"Vector databases allow us to attach arbitrary metadata to our vectors, enriching the user experience within the app."
"These are just flat vectors, we're not modifying them in any way."
"The direction here is important; the direction here is the direction of a vector that goes from the center to the vertex on the surface."
"So I like to think about vectors as more of like a design rather than just an image that you're making for a one-off case."
"These vectors V1 and V2 that are aligned with the semi-major and semi-minor axis of the ellipse are very special."
"The vector that doesn't change directions after linear transformation is an eigenvector of the matrix."
"Another way that this is described is that if you point your index finger on your right hand in the positive x direction, and your middle finger in the positive y direction, then raising your thumb points in the positive z direction."
"You can create any arrow of any length pointing in any angle through linear combinations of basis vectors."
"With RAG, you can scale to literally billions of vectors, which I think is pretty cool."
"Vectors are things that when you rotate them around by 360 degrees they come back to where you started, that's why vectors are spin one."
"They share the same vector but 4 is a multiple of the vector AB."
"If you extend your right hand in the direction of vector a then curve your fingers in the direction of B then your firm will go in the direction of the cross product."
"Changes in climate can increase the distribution or survival of disease reservoirs and vectors, leading to longer mosquito seasons and more days for infection."
"We want the vectors to carry some meaning."
"This is then going to generate other output vectors that are concatenated in order to generate a vector that actually has very good contextual awareness."
"So what it means to take the dot product between a vector and a by vector is to first project the vector onto the plane then rotate it 90 degrees and then either stretch or shrink it according to however much area is in B."
"now how do we know this is the correct way of representing a 4D space-time Vector as a two by two Matrix well for Pali vectors when we took the determinant we got the formula for the negative squared length of the vector."
"We can break up V into a portion that’s parallel with U, and perpendicular to U."
"The overall effect of transforming u to u double Prime by reflecting through V and then W is the same as rotating by two theta."
"Just pick any two vectors separated by half the angle you want to rotate by."
"The end result should be a rotation of that starting vector."
"The magnitude of the resultant when two vectors act in the same direction is equal to the sum of the individual magnitudes."
"Vectors are a very compact way of representing any type of data."
"Expression vectors are plasmids that carry cargo DNA into cells and allow the cargo DNA to be expressed."
"The projection of vector X onto vector V is given by X dot V over magnitude of V times u."
"If you want to find a vector from A to B, then AB is B minus A."
"The most important thing I can teach you is that for every vector, which is lots of things, any vector can be broken up into its components."
"Normalizing a vector means you're making the length of the vector 1, or in other words, you're making it a unit vector."
"So typically when you start in a second-year math course, you start with vectors in geometry, and that leads into sort of 3D geometry, line integrals, path integrals, all those sorts of things."
"If a vector is zero, it's zero in every coordinate system."
"That's why we express laws of nature in terms of vectors."
"Anyone in the world looking at this vector V equal to three, one can go ahead and draw that vector if they want to."
"If a vector has a space component equal to zero, that doesn't mean it's zero; it can have a time component."
"If we're given the dual vector, we know how to apply the inverse of the metric to it to give us the regular vector."
"Vectors that have the same or opposite direction but not necessarily the same magnitude are parallel vectors."
"Mosquitoes and ticks are vectors for many diseases that are potentially deadly."
"Vectors, I absolutely love vectors."
"A vector is just a thing with some direction and magnitude."
"Any vector in RN can be written as a combination of basis vectors with real number coefficients."
"Cosine similarity can measure how similar two vectors are."
"The magnitude of AB will be represented by the length of this line."
"Vectors have magnitude and direction, whereas scalars only have magnitude."
"The best way to learn about vectors is to actually apply them and work with them hands-on."
"The cosine similarity is essentially a score of how close two vectors are related."
"The contraction is the generalization of the inner product of two vectors."
"These two upward vectors add together to give a total upward vector of a hundred fifty feet per second at time of ejection."
"So you can visualize complex numbers as vectors in R2."
"Always think about what a vector means because that's the entire trick in this question."
"The tangent space is the space in which vectors exist."
"Vectors are just arrows and co vector fields are just level sets; these are all geometric objects that don't depend on any coordinate system."
"Your graph is a vector quantity, so you need to show the direction."
"The metric tensor, which takes two vector inputs, is sort of like a double Co vector."
"Think of the basis vectors as the building blocks used to create everything that belongs to that space."
"The vector \( \mathbf{K} \mathbf{a} \) is in the same direction with a magnitude of \( K \) times the magnitude of vector \( \mathbf{a} \)."
"A vector with a magnitude of \( 1 \) is a unit vector."
"The vector from \( A \) to \( B \) is \( B - A \), and the magnitude of a vector is the distance it travels."
"Parallel vectors are multiples of one another."
"If two vectors are parallel, it means the points they connect lie on a straight line."
"Nevertheless, you could add them and you could multiply by numbers, so we can call them vectors."
"It's a beautiful property of three dimensions that one is able to take two vectors and manufacture from them a third vector."
"In order for two vectors to be the same, they have to have the same length and the same direction."
"The cosine of the angle between the two vectors is given by the dot product of the two vectors divided by the product of the magnitudes."
"When you are adding together vectors, you literally just add the top number with the other top number."
"The position vector of a point means that the vectors are originating from their origin, O."
"Vectors are used in computer animation to determine the length of a shadow projected onto a flat surface."
"A set of vectors is said to be linearly independent if the only solution to their vector equation is the trivial solution."
"Two vectors v and w in R^n are orthogonal if and only if v dot w is equal to zero."
"V dot v is always non-negative and is zero if and only if v is equal to zero."
"We'll end up with the final vectors X that happen to be just really good at encapsulating context."
"Every vector W in V is uniquely expressed as a linear combination."
"The derivative of a rotating vector in an inertial frame is the derivative of that rotating vector in the rotating frame."
"Certain physical quantities have not only magnitude but direction as well; those quantities are represented by vectors."
"In physics, we're always adding vectors together."
"If you're ever asked to add vectors together or if you ever need to add vectors together, add them component by component."
"And you add the Y components of those vectors together, and that will be the Y component of the final resultant vector."
"No matter how many vectors you have... all you have to do is break every vector you have into X components and Y components."
"Vectors are useful for giving geometric representations of things like position, velocity, acceleration, momentum, and force."
"Vectors are seen as directed line segments with magnitude and direction."
"Vectors are not just numbers, they are directions and magnitudes combined."
"For every word, we could come up with a vector that encapsulates semantic information about the word."
"Vector quantities inherently have both magnitude and direction."
"The importance of vector quantities is that they provide more information than scalar quantities, especially when it comes to positioning of objects as they move through space."
"Dealing with vector quantities provides a lot more information in determining positioning in space and orientations in space for objects that are in motion."
"Any vector which is placed in a coordinate system as described can be presented as the sum of two mutually perpendicular vector components."
"When we talk about orientation of vectors in space, the standard procedure is to find the angle with respect to the positive x-axis in a counterclockwise direction."
"A useful way for us to describe the repeating pattern of the crystal lattice is through the use of vectors."
"If the vector doesn't change length, then its eigenvalue will be 1."
"If you remember the cross product of two vectors will give you a third vector, orthogonal to the other two."
"Once you have a basis, any vector in V can be written uniquely as an n-tuple."
"You must give both a size and a direction when you are talking about vectors."
"Perhaps what first years hate the most: three-dimensional vectors."
"We concluded that 2D vectors are actually pretty nice because they're very easy to look at and visualize."
"Remember, in 2D, if I had a force vector, we said that this can be split into two components: Fx and Fy."
"The magnitude is simply going to be Fx squared plus Fy squared added together and then square rooted."
"The magnitude of F is equal to the square root of Fx squared plus Fy squared plus Fz squared."
"If an inner product between two vectors is zero, it tells you they are orthogonal to each other."
"In order to determine the resultant vector of two force vectors, we use the parallelogram law or the triangle rule."
"It's really cool to think of these vectors as just being additions of other ideas."
"Each row of a matrix is simply a vector of numbers."
"An orthonormal basis means that you have a basis and all the vectors have unit length and are mutually orthogonal."
"Cross product would take two vectors and it would form a third vector... perpendicular to both of these vectors."
"The magnitude of a position vector is actually the distance between those two points, that shortest distance."
"Vectors are physical quantities defined by both a magnitude as well as a direction."
"The net horizontal displacement is 150 east and the net vertical displacement is 80 meters north."
"The angle between two vectors is defined by dividing their inner product by the product of their norms, which is a number between minus one and one."
"A vector is a quantity that has both a magnitude and a direction."
"The vector is always the same thing; it is an identity inside this vector space."
"A basis is a set of generators for the space of vectors."
"The vectors in B span the entire space and the vectors in B are also linearly independent."
"By definition, these three vectors are linearly independent."
"We define a vector simply as an arrow."
"The dot product takes two vectors down to a single number or a scalar."
"Functions f(x) ARE vectors. They are, mathematically speaking, the same as a geometrical vector."
"The unit vector is the vector itself divided by its magnitude."
"Expressing the vector in terms of its unit vector makes it much easier to compute the direction cosines."
"A vector is a directed line segment with a direction or orientation and a length or magnitude."
"The magnitude of a vector is always positive."
"Vectors lie on a line, and that line is called the line of action."
"That's another real-life application of trig; it helps you to solve physics problems with vectors and their components."
"It will turn everything into embeddings and all of the vectors that you might need."
"There are 3 vectors that 120 degrees apart and they're rotating."
"This idea of representing words as vectors turns out to be incredibly useful and powerful."
"Vectors are quantities that have magnitude and direction."
"Multiplying a vector by a scalar changes the vector's magnitude but not its direction."
"Vectors are infinitely scalable and they just look really, really amazing."
"The concept of a vector does not depend on an arrow anymore than the concept of a number depends on a length."
"The set of free vectors \( \mathbf{V} \) in \( \mathbb{R}^3 \) they form a linear space."
"Two vectors are said to be perpendicular to each other, or normal to each other, or orthogonal to each other, when the dot product between the two vanishes."
"Quantities that require both magnitude as well as direction are called vectors."
"We started by fitting the text or the vectors between two curves."