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Say you have vectors p0 and p1. The midpoint between them is simply (p0+p1)/2. More generally, the line segment defined by p0 and p1 can be generated by varying t between 0 and 1 in the following linear interpolation:

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Because the dot product between a and b is also defined as |a||b|cos(alpha) (where alpha is the angle between the two), the result will be 0 if the two vectors are perpendicular, positive if the angle between them is less than 90, and negative if greater. This can be used to tell if two vectors point in the same general direction.

Say that you have a list of directions represented as unit vectors, and you want to find which of them is the closest to dir. Simply find the largest dot product between dir and a vector in the list. Likewise, the smallest dot product will be the direction farthest away.

Say that you have an arbitrary point in space, p0, and a direction (unit) vector, dir. Imagine that an infinite line goes by p0, perpendicular to dir, dividing the plane in two, the half-plane that dir points to, and the half-plane that it does not point to. How do I tell whether a point p is in the side pointed to by dir? Remember that dot product is positive when the angle between vectors is less than 90 degrees, so just project and check against that:

Intersection of a line and a plane

A vector is a mathematical way of representing a point. A vector is 3 numbers, usually called $x$, $y$ and $z$. You can think of these numbers as how far you have to go in 3 different directions to get to a point. For instance, put one arm out pointing to the right, and the other pointing straight forward. I can now give you a vector and you'll be able to findthe point I'm talking about. For instance, if I say $x=3$, $y=1$, $z=5$, you find the point by walking 3 metres in the direction of your right hand, then 1 metre in the direction of your left hand, and then getting a ladder and climbing up 5 metres. Here is a picture of a vector.

Picture of a vector and directions

Vectors are written as $(x,y,z)$, for instance $(1,2,3)$ means move 1 in the x-direction, 2 in the y-direction and 3 in the z-direction.

One confusing thing about vectors is that they are sometimes used to represent a point, and sometimes they are used to represent a direction. The vector $(1,0,0)$ can mean both "the point you get to if you move 1 unit in the x-direction from the starting point'', or it can mean "move 1 unit in the x-direction from where you are now''.

If I start at the point represented by the vector $(0,0,0)$, then I travel along the vector $(1,2,3)$, and then I travel along the vector $(4,5,6)$, where will I be? This vector is called the sum of the two vectors, $(1,2,3)+(4,5,6)=(?,?,?)$.

If I translate the vector $(1,2,3)$ by 5 in the y-direction, what is the vector afterwards? In general, if we write a translation as $T$ and a vector as $\mathbf{v}$, then the new vector if written as $T\mathbf{v}$. For example, if $T$ is ``translate 1 unit in the x-direction'' and $\mathbf{v}=(0,0,0)$, then $T\mathbf{v}=(1,0,0)$.

The answer is $(1,2,3)+(4,5,6)=(5,7,9)$. This is because $1+4=5$, $2+5=7$ and $3+6=9$. The reason the answer is those sums is because to find out how far we go in the x-direction, we work out how far we have gone in the x-direction from the first vector, and add on how far we have gone in the x-direction from the second vector. We do the same for the y- and z-directions. In general,$(x,y,z)+(u,v,w)=(x+u,y+v,z+w)$.

The answer is $(1,7,3)$ because we add the vector $(0,5,0)$ to move 5 in the y-direction, and $(1,2,3)+(0,5,0)=(1,7,3)$. In general, a translation $T$ is something like ``add $(u,v,w)$ to the vector'', to $T(x,y,z)=(x,y,z)+(u,v,w)=(x+u,y+v,z+w)$. If you haven't done matrices and vectors at school, ignore the next bit.

Because the x- and z-directions are 0 in all the vectors, we can ignore them and just look at the y-directions. At time $t=0$, the stone is at position 0, with velocity 4 (with vectors, the stone is at position $(0,0,0)$ with velocity $(0,4,0)$). Therefore, at time $t=0.2$, the stone will be at position $0+4 \times 0.2=0.8$. At time $t=0.2$, the velocity will now be $4+0.2 \times -9.81 =2.038$. So, at time $t=0.4$, the stone will be at position $0.8+2.038 \times 0.2=1.2076$. At time $t=0.4$, the velocity will be $4+0.4 \times -9.81=0.076$. So at time $t=0.6$, the stone will be at $1.2076+0.076 \times 0.2 = 1.2228$. At $t=0.6$, the velocity will be $4+0.6 \times -9.81 = -1.886$, so at $t=0.8$ the position will be $1.228-1.886 \times 0.2 = 0.8508$. At $t=0.8$ the velocity will be$4+0.8 \times -9.81 = -3.848$, so at $t=1.0$ the position will be $0.8508-3.848 \times 0.2 = 0.0812$. So, after 1 second, the position of the stone will be $(0,0.0812,0)$.

Whereas the Optical Flow Accelerator accurately tracks pixel level effects such as reflections, DLSS 3 also uses game engine motion vectors to precisely track the movement of geometry in the scene. In the example below, game motion vectors accurately track the movement of the road moving past the motorcyclist, but not their shadow. Generating frames using engine motion vectors alone would result in visual anomalies like stuttering on the shadow.

For each pixel, the DLSS Frame Generation AI network decides how to use information from the game motion vectors, the optical flow field, and the sequential game frames to create intermediate frames. By using both engine motion vectors and optical flow to track motion, the DLSS Frame Generation network is able to accurately reconstruct both geometry and effects, as seen in the picture below.

Unlike pixel-based designs, vector art boasts a smaller image file size while ensuring superior quality, whether enlarging or zooming out on images. The unique trait of maintaining high resolution regardless of manipulation speaks volumes about its versatility in video game art and design. This has prompted game designers to embrace vector art in crafting the artistic elements of 2D game styles.

The complexity of the game art style. The more intricate and detailed the 2D art style, the more resources it demands. For instance, a detailed style like vector art necessitates a larger budget and longer timeline than a simpler pixel art style. The reason? The increased workforce and the specialist skills needed to create top-notch vector art.

Vector 2 um jogo de plataformas 2D onde voc tem de tentar escapar de uma instalao de pesquisa repleta de armadilhas mortferas, lasers, minas, elevadores... tudo em seu redor est desenhado para trabalhar contra voc.

Os controles so exatamente os mesmos do primeiro jogo na saga. Sua personagem corre para a frente sem parar. Basta deslizar para cima para saltar e para baixo para rebolar sob as coisas. Tambm, por vezes, voc pode deslizar seu dedo para a frente para fazer com que sua personagem corra.

When considering the risks associated with Office 365, email security is a vital consideration. However, email is not the only attack vector within the Office 365 product suite. Some of the most common security risks that Office 365 users encounter include:

Office 365 is a major asset for many organizations, especially when remote work makes the ability to communicate and collaborate online more important than ever. However, Office 365 also creates a number of potential attack vectors as cybercriminals abuse these same features.

Protecting against the cybersecurity risks associated with Office 365 requires a multi-layered security solution. At the network level, an organization should implement solutions for inspecting emails and other shared content for malware and phishing content, attempted data exfiltration and other threats. However, it is possible that some attacks may slip past these network-level defenses. This makes a comprehensive solution that secures users, devices, and access necessary to ensure that all potential attack vectors are closed and that an attack can be detected and remediated at any stage of its lifecycle.

The Check Point researchers were able to uncover the attack vector which affects major websites and social networks worldwide, including Facebook and LinkedIn. Check Point updated Facebook & LinkedIn of the attack vector early in September.

A detailed and technical disclosure of the attack vector will be published by Check Point only after the remediation of the vulnerability in the major affected websites, in order to prevent attackers from taking advantage of this information. ff782bc1db