# Tag Info

## New answers tagged vectors

1

Here is how to solve these problems in general. Make a sketch for the balance of forces. and using trigonometry write down the $x$ and $y$ components of the vectors $$F_{BC}\cos(\varphi)+F_{BD}\cos(\theta) = F \\ F_{BC}\sin(\varphi)-F_{BD}\sin(\theta) = 0$$ Now solve for $F_{BD}$ and $F_{BC}$.

1

I wonder if you're getting mixed up with propagation of waves in a physical medium like a string. If you have a wave travelling on a string then it has a velocity along the string, but the string is also oscillating normal to its length. So if you stretched the string along the $z$ axis, as the wave travelled along the string (i.e. the $z$ axis) the string ...

0

A photon is the quantized unit of the electromagnetic field. If you have en electromagnetic wave propagating in the x-direction, this must consist of a magnetic field and an electric field oscillating perpendicularly to the direction of travel, and to each other, i.e. in the y and z directions. If you have a wave with a frequency of, as an example, 50Hz, it ...

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You're confusing the process of quantization with the wave-nature of propagating electromagnetic fields. When you look at a Electromagnetic waves as photons, this means you don't look at their wave-characteristics, and you consider them as particles travelling with the speed of light, and those particle could "hit" electrons and knock them our of the atom ...

1

Tensors (or rather tensor fields in case of differential geometry) are very generic and not particularly intuitive objects that can fill a lot of roles - volume elements, endomorphisms, Riemannian metrics are just a few things you can describe with tensors. However, to get an intuition about co- and contravariance, it's enough to look at tangent vectors and ...

1

Although this is saying the same thing as Lionel's answer and Mark's answer from a different standpoint, another idea that I like in describing the tangent space is to think of the one dimensional $C^1$ space curve (or spacetime curve) within the manifold $M$ as a grounding concept. So our fundamental idea is some function ("A Path" or "A Trail") through the ...

3

This whole business of covariant vs contravariant is very old school. Some very old texts go into ways of visualizing this. I would suggest instead learning about tangent vectors (contravariant) and 1-forms (covariant) and the equivalence between tangent vectors and directional derivatives. Associate the vector $\vec{v}$ with the derivative operator ...

2

Here is a visualization from Geometrical Methods of Mathematical Physics by Schutz. The co-vector is here called a "one form". His notation $\langle \tilde{\omega} , \bar{V} \rangle$ is equivalent to $\omega_\alpha V^\alpha$, which you might be used to seeing. Note that when the magnitude of $\bar{V}$ increases, the arrow gets longer. When the magnitude ...

1

Changing the "wind" to a "current" in an attempt to save the question, the OP's solution is still not quite correct. By subtracting the current's west component from the boat's overall, west velocity over the ground, the OP has correctly found the west component of the boat's velocity through the water. However, he has forgotten that the boat must cancel ...

3

This question can't be answered because important information is not given. We have no idea how much influence, if any, the wind has on the boat's velocity. In the limiting case, if the boat's wind resistance is 0, then the wind has no effect. The question also asks what the boat would do in still water, but we were never told how still or not the water ...

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How do we prove that any directions are orthogonal? [...] we can use the pythagorean theorem. This involves of course a definition of (how to measure or compare) "angle(s)" in the first place; such that one may comprehend statements about (distinct) angles being "equal" (or else: "not equal") for instance in Euclid's 4th axiom (on "right angles") or in ...

3

The explanation does not lie in the vectorial nature of the quantities at hand, but rather in the fact that they can be viewed as functions of several variables. Just as a scalar function $f$ can depend on a variable $x$ and be denoted $f(x)$, or it can depend on two variables $x$ and $y$ and be denoted $f(x,y)$, and similar for more variables, a vector can ...

4

It depends on how you define orthogonality, or, as OSE puts it in his comment, "Orthogonality is usually tested using some defined inner product." I'll expand on this a bit. In order to mathematically answer the question Is direction A orthogonal to direction B? we need a definition of the terms "direction" and "orthogonal." The standard ...

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Since you are probably not ready to simulate the response of a steering column or wheels right away, I suggest the following as a warm-up. To get the ball rolling, as they say. To a large extent, steering doesn't impact the speed of the vehicle, so you can start with an acceleration vector that is orthogonal to the velocity vector and has magnitude ...

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