New answers tagged vectors
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What exactly is happening to acceleration when direction changes?
very good question.
we know that speed is a vector. for changing a vector, it needs to be added to another vector.
this question, is a very good example to talk about steady state and transient state.
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What exactly is happening to acceleration when direction changes?
Direction of motion changes due to acceleration but there is not any compulsion that there should or should not be any change in acceleration when direction changes.
For example when we throw a object ...
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What is meant by 'The components of a force along a given axis'?
Say you know a force of magnitude $F_a$ acts along the direction $\hat{a}$
The force vector is thus $$\vec{F} = F_a\, \hat{a}$$
In reverse, to find out the magnitude, you can use the dot product
$$ ...
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What is a Tensor, intuitively?
Before we understand a tensor, we need to understand what a co-vector is. A co-vector is essentially a linear map from a vector to a number. A simple example of such is the gradient covector. For ...
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How can this be a vector?
The vector of Pauli matrices is $\boldsymbol \sigma = (\sigma_{x}, \sigma_{y}, \sigma_{z})$ and therefore
\begin{equation}
\boldsymbol m = \begin{pmatrix}
\langle \psi| \sigma_{x} | \psi \rangle \\
\...
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What is a Tensor, intuitively?
I'll flip the order of questions.
What makes a tensor a tensor?
A tensor is indeed something that transforms as a tensor. No further definition is needed. An example might help. The metric tensor is ...
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Representing vectors in non-inertial rotating frame of reference
Consider two frames of reference, O and O*. O is the origin of a fixed point in an inertial frame. O* is the origin of a fixed point in a noninertial frame. In general O* can be accelerating with ...
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What direction should i exactly put for negative displacements?
Displacement is a vector. A vector has a direction, not a sign.
It is frequently convenient to choose a coordinate system where vectors to the east are represented with positive numbers and vectors to ...
rob♦
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What direction should i exactly put for negative displacements?
You define a positive x axis, positive y axis, positive z axis.
E.g -1,0,1
A displacement to the left in this case, is denoted by a negative, and displacement to the right is denoted by a positive.
By ...
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What direction should i exactly put for negative displacements?
Start by defining a unit vector (positive direction) eg $\hat e $(ast) which in this case might be to the right in the diagram below and $\hat w$(est) to the left.
This is just equivalent to using the ...
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What direction should i exactly put for negative displacements?
If you are considering the displacement from C to d, then yes, this is a negative displacement. However, if you are considering the total displacement (i.e. the one defined with A being the initial ...
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Change in kinetic energy when force is perpendicular to velocity
It is said that when force is applied perpendicular to velocity, there is no change in kinetic energy since there is no change in speed.
Exactly what happens when there is uniform circular motion.
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Change in kinetic energy when force is perpendicular to velocity
The perpendicular component of a force will never change the speed. This is true.
What is happening in your case is at the instant that the force along the y-axis is applied, it will, for that short, ...
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Is there an official name for "Lorentz Pairs" like energy and momentum?
What you might be looking for is the concept of Lorentz covariance. A Lorentz covariant quantity is a (finite collection of) quantities which are taken to linear combinations of themselves under ...
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How to remember which way the magnetic field goes?
The names "north pole" and "south pole" for the ends of a magnet were introduced a long time ago, well before anyone understood how magnetism arises and how it relates to electric ...
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In an $n$ particle system, why is the Hamiltonian summed over $n$?
The index $i=1,\ldots n$ is a particle index; not a coordinate index. The $i$th particle carries a 3-momentum $p_i\in\mathbb{R}^3$.
In the Hamiltonian $p_i^2=p_i\cdot p_i$ is a dot/scalar product.
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In an $n$ particle system, why is the Hamiltonian summed over $n$?
The hamiltonian is a scalar quantity as it represents total energy of a system.
Each particle momenta in your equation $(1)$ is the magnitude of said particle's momenta $p_i = \sqrt{\sum\limits_i^d \...
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The meaning of negative sign in vectors?
By convention, the magnitude of a vector is always of a non-negative number. So for a vector $\vec{a} = (-5) \hat{\imath}$ the interpretation is as follows
$$ \vec{a}\;\;\; = \underbrace{5}_\text{...
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The meaning of negative sign in vectors?
In rectangular coordinates, the general equation for the velocity (or any) vector is
$$\vec v=v_{x}\hat i+v_{y}\hat j+v_{z}\hat k$$ where the values $v_x$, $v_y$ and $v_{z}$ are real numbers that can ...
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Is the space-part of a four-vector temporally connected to the time-part and vice-versa?
I offer up a different view that does tie space and time together as you are thinking, based on Loop Quantum Gravity.
LQG is an alternative theory of space (alternative to string theory) that is ...
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Is the space-part of a four-vector temporally connected to the time-part and vice-versa?
Can we say that it's the time part that's involved in defining the space part and the space part in defining the time part?
A displacement 4-vector has
a time-component that has nothing to do with ...
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What specifically about the torque vector is perpendicular? Is the torque vector like this only so that it works smoothly with linear algebra?
It's neither calculus nor linear algebra; rather, it's geometry, or perhaps representation theory, or Clifford algebra... each of which can lead to a long an in-depth answer. That will not be ...
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Notation for contracting vectors using metric tensors
If we take a vector $A$, which has three components, my understanding is that we can write this using Einstein notation as $A_{u}$ where this is actually $A_1+A_2+A_3$.
No. $A_\mu$ is the $\mu^{th}$ ...
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Clarifications on proving lightlike vectors must be orthogonal with themselves
We say two vectors $v,w$ are orthogonal with respect to $g$ if $g(v,w)=0$. That's just by definition. In components, this is written as $g_{ab}v^aw^b=0$. If you're working in Minkowski with standard ...
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Abstract definition of four-vector
95% correct, let me add some elements.
We define the "mathematical Minkowski spacetime" as the vector space $\mathbb R^4$ endowded with a bilinear form we denote by $\eta$ ("mostly ...
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Why do coordinates have to be inverted to form a dual basis?
I'm not sure if there's a typo in your question, but the basis vectors induced by the coordinates $(u,v,w)$ are e.g.
$$\hat e_u = \frac{\partial x}{\partial u} \hat i + \frac{\partial y}{\partial u} \...
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How time-like unit four-vector is tangent to the observer's world-line?
I don't get how Time like unit four vector is tangent to observer's world line!!
It is not necessary for the timelike unit four-vector to be tangent to an observer's worldline. However, since we have ...
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Do we have to use the metric constructed from the tangent basis to form a line element?
If I understand the question properly, the idea is the following. We first consider Cartesian coordinates $(x,y,z)$ which induce the orthonormal coordinate basis vectors $\hat e_x,\hat e_y,\hat e_z$. ...
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Simple difference between module of velocity and time derivative of module of position
The first describes the rate at which the distance between the object and the (often arbitrary) origin is changing, whereas the second is the actual speed of the object (the speed being the magnitude ...
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Which force should I break into components?
You are welcome to resolve the normal force into horizontal and vertical components if you'd like. If you do so, this would be the result:
Newton's 2nd law then takes the form
$$\sum F_x = m a_x \...
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Which force should I break into components?
But what if I equated gravity to be equal to the component of the
normal force in the upright y axis (in the direction of gravity. )
The origin of the gravitational force is not the component of the ...
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Which force should I break into components?
The second approach is wrong. The normal force is normal to the surface on which the block rests. Since there is no acceleration in direction normal to the wedge surface:
$$\sum F_{normal} = N-mg\cos{...
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Why cannot we add a scalar to a vector of the same dimensions?
This question is old but I'll answer it for those who might be wondering if this is possible
You can't.
$1≠ [1, 0, 0]≠\begin{pmatrix}1 & 0 & 0\\\ 0 & 1 & 0 \\\ 0 & 0 & 1 \end{...
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