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You can also see it in the context of group representation theory. $O(n)$ is generated by the same infinitesimal generators as $SO(n)$ plus the discrete generator $\mathbf P$ which is parity. The parity operator generates a discrete subgroup $\mathbb Z/2\mathbb Z$ in $O(n)$). So if we choose a representation $\rho_{SO}$ of $SO(n)$ and a representation ...

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First of all, as this is a homework question, I can't tell you the complete solution. Choose P as the origin of coordinate system and resolve the forces into x and y component. And as the body is in equilibrium, the net force is zero. So you get these two relations (when the net force on x and y component equated to zero.) $$G\cos\theta=H\cos\phi\\ ... 0 A vector is a basic mathematical construct. There can be many types of vectors (velocity, position, force, etc...). Your first chapter is defining a vector in general. The second is introducing a vector that describes position. If you take the difference of two position vectors, it is a displacement vector. Thus. Through algebra the difference of a ... 2 This is one of those things that (intentionally) gets conflated, though it may be better if we were more consistent about keeping them separate. So, points don't form a vector space. It makes no sense to ask "what's the location of New York plus the location of DC". However, given two points we can subtract them and get a displacement, and we can add that ... 3 This is an interesting question and the answers which have been given show that the v in your equation should be called the magnitude of the velocity or just the speed of the wave. The mixing of the terms speed and velocity happens all the time. Now there is an equation for wave velocity but in comes about in a somewhat convoluted way. Suppose that you ... 0 Since the wavelength is the distance between the two consecutive crests or troughs it isn't a vector quantity. It is scalar. Simillarly, Frequency is a scalar quantity, since it just number of crest and trough per second. How can such number have direction. 0 Vectors and vector notation can tend to simplify equations and algebra. They might feel like they just complicate things at first, but once you get used to them they can provide an intuitive way to simplify calculations and concepts. Consider for example Newton's second law. While without vectors we'd have to write$$F_x = m \times a_xF_y = m \times ...

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What would your "single variable" for direction look like? You'd need two angles, or something equivalent, to describe it, and two angles and a magnitude is exactly a vector in spherical coordinates.

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Always, always, always start problems like this by drawing a diagram: This make it obvious why cos and sin are used as they are.

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By convention, North and South rank higher in the English language. Thus we have Northeast and not Eastnorth. You have presumed a flat earth. If you start 1.5 km from the North Pole, then after skiing 3 km east and 1.5 km north, you will be at the North Pole, 1.5 km North from your starting point.

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