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Jacobian and Hessian of a vector

As of the first part of your question, the Jacobian you have calculated is indeed correct. As of the second part, I would like to note, first, that Hessian is typically defined for scalar-valued ...
Egor Larionov's user avatar
1 vote

Find distance between 3d point and 3d curve

The curve you is $ p(t) = p_0 + t p_1 + t^2 p_2 $ And you want the value of $t$ when $p(t)$ is closest to a point $q$. At the minimum, you will have $ p'(t) \cdot (p(t) - q) = 0 $ That is, $ (p_1 + 2 ...
c'est pas normale's user avatar
0 votes

How to find angle for area of triangle

In $\triangle ODB$ we have $OD=1, OB=\sqrt{2}, BD=\sqrt{3}$. Since $OQ \perp BD$, $\triangle OQD \sim \triangle OBD \implies \frac{OQ}{OD}=\frac{OB}{BD}, OQ=\sqrt{\frac{2}{3}}$. Now use right triangle ...
Vasili's user avatar
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How to find angle for area of triangle

Find the equation of the plane $\Pi_3$. To find this, we have the direction ratio of the plane given in the question. Find the equation. Then find $P$ and $Q$. And prove what's required. Don't solve ...
MathStackexchangeIsMarvellous's user avatar
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How to treat juxtaposition between vector and column and how to handle them?

If $s,t \in F^n$ are column vectors; i.e., we can write $$ s = \begin{pmatrix} s_1 \\ \vdots \\ s_n \end{pmatrix} \text{ and } t = \begin{pmatrix} t_1 \\ \vdots \\ t_n \end{pmatrix},$$ then they are ...
Dheeran Wiggins's user avatar
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Proof of $\mathbf{E}^*\times (\nabla\times \mathbf{E}) =\mathbf{E}^*.(\nabla)\mathbf{E}+\frac{1}{2}\nabla \times \mathbf{E}^*\times \mathbf{E}$

Ok got it (thanks J.G and Mathematica) $\mathrm{Im}\left(E^*\times(\nabla\times E\right)= \frac{1}{2}\mathrm{Im}\left(E^*\times E\right) +\mathrm{Im}\left\{E^*.(\nabla) E\right\} +\mathrm{Im}\left\{(\...
Cuki79's user avatar
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1 vote
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Is it possible to reflect a line over another line in three-dimensional space if both lines are skew lines?

Suppose you want to reflect the line whose parametric equation is $p(t) = p_0 + t \ u $ about the line whose parametric equation is $ q(s) = q_0 + s \ v $ Reflection about $q(s)$ is equivalent to ...
c'est pas normale's user avatar
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What are the implications of representing basis vectors as directional derivatives?

Math perspective here. In differential geometry, the tangent space $T_p M$ at a point to a differentiable manifold (like spacetime) may literally defined to be the $\mathbb{R}$ vector space of ...
Integral fan's user avatar
2 votes

Why is the magnitude of the cross product equal to the parallelogram spanned by the two vectors?

Regarding the second question: The formula for $|a\times b|$ follows from four axioms: $|a\times b| = |a||b|$ if $a\perp b$. $a\times a = 0$. Bilinearity of $\times$. $a\cdot b = |a||b|\cos\theta$. ...
Nicholas Todoroff's user avatar
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Newton-Raphson algorithm proof: derivating a matrix

$ \def\bb{\mathbb} \def\d{\cdot} \def\h{\tfrac12} \def\f{f_+} \def\o{{\tt1}} \def\p{\partial} \def\H{{\large\cal H}} \def\bR#1{\Big(#1\Big)} \def\BR#1{\Big[#1\Big]} \def\LR#1{\left(#1\right)} \def\...
greg's user avatar
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3 votes

Why is the magnitude of the cross product equal to the parallelogram spanned by the two vectors?

Q1. What is the purpose of wanting the cross product to have magnitude equal to the parallelogram spanned by $a$ and $b$? I'm not sure what sort of answer you expect to get to this question. The ...
Blitzer's user avatar
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0 votes

Properties of 3D Vectors and Lines, and Planes

Let's say that $\vec{a} = (1,0,0)$, $\vec{b} = (0,1,0)$ and $c=(-1,0,0)$, then you can draw all lines $l_1$, $l_2$ and $l_3$ in the same $2D$-plane (the $z$-coordinate never changes from zero to ...
Dominique's user avatar
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2 votes
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Discrepancy in answer to a question about vectors

For me the correct answer is: those conditions are impossible. In other words: there is no such value of $n$. Let's follow your second attempt, but I will use matrix notation for rotating a vector by ...
Rupert Rybka's user avatar
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How can I make a vector of the products of the elements $(a_0b_0,a_0b_1,a_1b_0,a_1b_1)$ from the vectors $(a_0,a_1)$, $(b_0,b_1)$?

You can do $$\begin{pmatrix}a_0\\ a_1 \end{pmatrix}\begin{pmatrix}b_0&b_1\end{pmatrix}=\begin{pmatrix}a_0b_0&a_0b_1\\a_1b_0&a_1b_1\end{pmatrix}$$ and then maybe list the rows together into ...
ultralegend5385's user avatar
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Frames of reference in Coriolis' equation

The vector quantity $\vec{r}$ should be expressed in the rotating frame. Check https://en.wikipedia.org/wiki/Rotating_reference_frame#Time_derivatives_in_the_two_frames.
JorgeOvi's user avatar
1 vote

Find the plane passing through $A$ and tangent with a sphere $(S)$ such that the distance between $B$ and this plane is greatest.

Drawing an accurate sketch of the sphere and points (see below) can help to find the solution without so many calculations. In fact, it looks like the furthest intersection between line $OB$ and the ...
Intelligenti pauca's user avatar
1 vote
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How to know whether a set of points can be rotated to lie in positive orthant?

I think I have, many years later, stumbled on an answer to this question. Stack the set of $d$ points into a matrix $X \in \mathbb{R}^{N\times d}$. If the set of points can be rotated into the ...
Will Dorrell's user avatar
2 votes
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Find the plane passing through $A$ and tangent with a sphere $(S)$ such that the distance between $B$ and this plane is greatest.

Edit : your approach (considering an adequate slicing plane) is a good one, very "geometric" in its essence. Here is an alternative solution, analytical, using Lagrange multipliers (see ...
Jean Marie's user avatar
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1 vote

Solving vector equations in $ℝ^3$

$ \newcommand\R{\mathbb R} \newcommand\form[1]{\langle#1\rangle} \newcommand\lcontr{\mathbin\rfloor} \newcommand\rcontr{\mathbin\lfloor} $Here is an approach for (1) using geometric algebra that does ...
Nicholas Todoroff's user avatar
1 vote
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Need help in using the right-hand rule for cross product

One way to find $\vec{MN}$ is to take the two candidates (each in the opposite direction of the other) and check to see which one is in the direction from $M$ to $N.$ Consider the plane $\pi_2$ ...
David K's user avatar
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1 vote
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Solving vector equations in $ℝ^3$

(i) We can write $$\vec{a} \times[(\vec{r}-\vec{b})\times\vec{a}]=|\vec a|^2\vec r-(\vec a\cdot\vec r)\vec a-|\vec a|^2\vec b+(\vec a\cdot\vec b)\vec a$$ $$\vec{b} \times[(\vec{r}-\vec{c})\times\vec{b}...
mathlove's user avatar
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Find the values of $ t$ so that the tangent line to the given curve contains the given point

As themathandlanguagetutor said, the equation of the tangent line to the curve at a fixed time $t$ is given by: $$L(s) = r(t) + r'(t) s, \ s \in \mathbb{R}$$. So first let's write the equation of the ...
Sophia 's user avatar
1 vote

Solving vector equations in $ℝ^3$

(i) The provided answer is wrong. For instance, if $\vec{a}=(1,1,0)$, $\vec{b}=(0,1,0)$, $\vec{c}=(0,0,1)$, and $\vec{r}=\frac{\vec{a}+\vec{b}+\vec{c}}2=(1/2,1,1/2)$ then Mathcad yields $$\vec{a} \...
Alex Ravsky's user avatar
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0 votes

Deriving the Maximum Range of a Particle With A Constant Force Using Geometric Algebra

Equation 2.8 reads $$r = \frac{2\left(\vec{g}\wedge\vec{v}_0\right)\cdot\left(\vec{v}_0\wedge\hat{r}\right)}{\lvert \vec{g}\wedge\hat{r}\rvert^2} = \frac{2v_0^2}{g}\frac{\left(\hat{g}\wedge\hat{v}_0\...
Timothy Wofford's user avatar
1 vote

How to calculate angle between vectors in a way that is invariant under translation?

There are many things that can be called "vectors" in mathematics, so it's important to be clear about what kind of vector you mean when you ask about "vectors." Some vectors do ...
David K's user avatar
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1 vote

How to calculate angle between vectors in a way that is invariant under translation?

If you shift your vectors, you get new vectors: you should always look at them from the original point. The new vector will have a different angle between them. If you want to calculate the original ...
MikhailSemenov's user avatar
0 votes

Need help in explaining weirdness when representing complex numbers using vectors

You found out that if we have two complex numbers $z_{1}= (a + bi) $ and $z_{2}= (c+di)$ that the product $ac - bd + ( bc +ad)i $ is not what you want for the dot product. You would like $ac + ...
kirk beatty's user avatar
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How are the tangential and normal components of the acceleration vector derived?

Let's first have a look at the geometry representation of the vectors The accelerator vector $\vec{a}$ could be broken down into 2 components: $a_{N}\vec{N}$ and $a_{T}\vec{T}$. $\vec{T}$ is the unit ...
Gwen's user avatar
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1 vote
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Euclidean distance between 4-dimensional vectors

View each row of your vector as a dimension. Subtract one vector from the other, so you have a set of lengths for each dimension. The case for the first 2 dimensions is the classic case, yielding ...
RobinSparrow's user avatar
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0 votes

Proof that two matrices are row-equivalent iff they have the same nullspace

Let $A, B$ be the two matrices, that define linear maps from the space $E$ and $F$. Let $(e_1, \dots e_l, e_{l+1}, \dots, e_n)$ be a basis of $E$ where $(e_1, \dots e_l)$ is a basis of the shared null ...
mathcounterexamples.net's user avatar
1 vote
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A trivial problem about two identical hyperplanes

Your problem has a positive answer. In fact much more is true. Let $X$ be a real vector space $\not=\{0\}$ with an inner product $\langle~,~\rangle$. Moreover for $0\not=a\in X$ and $b\in\mathbb{R}$ ...
Jens Schwaiger's user avatar
3 votes

Need help in explaining weirdness when representing complex numbers using vectors

I prefer to discuss the issue differently. Given any complex number $~z \in \Bbb{C},~$ you can uniquely identify $~x,y \in \Bbb{R},~$ such that $~z = x + iy ~: ~i = \displaystyle \sqrt{-1}.$ Note ...
user2661923's user avatar
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10 votes
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Need help in explaining weirdness when representing complex numbers using vectors

The heart of your issue is this: the dot product of two complex numbers and the usual product of complex numbers is not the same product. Full stop. There is no "in some cases it's the same and ...
Vercassivelaunos's user avatar
2 votes

Need help in explaining weirdness when representing complex numbers using vectors

Your Inconsistency When you calculate the magnitude $$ \left|\vec{OM}\cdot\vec{OM}\right| = 3^{2} + 4^{2} $$ you multiply the real part of the vector with the real part, the imaginary part with the ...
acat3's user avatar
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1 vote

How to Calculate the Curl for $\vec{\nabla}\times\hat{a}$ for unit vector $\hat{a}$ in Cartesian, Cylindrical, and Spherical Coordinates

Or simply $\nabla\times\mathbf{a}=\epsilon_{ijk}\partial_ja_k$. I suppose that Levi-Civita tensors can be used to split the metric basis. For a general vector coordinate $\mathbf{x}=x\hat{\textbf{e}}...
MathArt's user avatar
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0 votes

How to find the shortest line segment connecting two skew lines, with constraint that line is parallel to some plane?

Taking the derivative of the distance between two points on a translating plane was not as hard as I feared. For two lines, each defined by a point on the xy plane and a direction vector with z ...
BitPusher16's user avatar
1 vote
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Find all possible values of a parallel vector with equal magnitude

Since $u$ and $v$ are parallel vectors, we can write $v=t\times u$ for some scalar $t$. So we have $ai-bj+ck=ti-8tj+4tk$. Equating components gives $a=t,b=8t,c=4t$. Since the length of $u$ is 3, we ...
Red Five's user avatar
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2 votes
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Identifying the centroid, incenter, circumcenter, and orthocenter of a triangle from a set of vector equations

To save typing and reduce confusion over subscripts, I'll define $A:=A_1$, $B:=A_2$, $C:=A_3$, and $P:=P_1$, $Q:=P_2$, $R:=P_3$, $S:=P_4$. (I'm ignoring the usage of these labels in the "Columns&...
Blue's user avatar
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0 votes

Find all possible values of a parallel vector with equal magnitude

Well, if two vectors are parallel their cross product is zero. There should only be one possible set of values for a vector with a fixed magnitude in 3 dimensions to be parallel to another one unless ...
mike1994's user avatar
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4 votes
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Shortest path on the surface of a cylinder between given points $A$ and $B$

You're right. The shortest path is not planar. It's part of a helix. Think about even a very short trip along the thread of a bolt. This is from (my 1959 edition of) Hugo Steinhaus' Mathematical ...
Ethan Bolker's user avatar
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1 vote

How to find the shortest line segment connecting two skew lines, with constraint that line is parallel to some plane?

Let the two lines be $ p_1(t) = P + t d_1 \tag{1}$ $ p_2(s) = Q + s d_2 \tag{2}$ where $d_1, d_2$ are the direction vectors of the two lines respectively. The vector connecting $p_1(t)$ to $p_2(s)$ is ...
c'est pas normale's user avatar
1 vote
Accepted

a Cauchy Schwarz application

Define $x_i^2 = a_i^2+b_i^2, x_i >0$, then Using Multinomial theorem, $$ \left(x_1+x_2+\cdots+x_n\right)^2=\sum_{ \substack{ 0 \le j_1, j_2, \ldots, j_n \le 2\\ j_1+j_2+\cdots +j_n = 2}} \frac{2}{...
Sam's user avatar
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0 votes

Advanced Algebra Problem Maybe linked with Vectors?

The triangle you formed by substruction of pairs of vectors has sides $5,7,8$. By Brahmagupta's formula its area is $10\sqrt3$. On the other hand, the area of the triangle is the sum of the areas of ...
Bob Dobbs's user avatar
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Advanced Algebra Problem Maybe linked with Vectors?

Taking the resultant of the polynomials, with $r=(x+y+z)^2-100$, one obtains \begin{align*} r&=-91,\quad x=0,y=5, z=-8,\\ r & = 29,\quad 8y=5x,5z=8x, 129x^2 = 1600. \end{align*}
Dietrich Burde's user avatar
1 vote
Accepted

Derivation of Continuity Equation for an Incompressible flow

For an arbitrary scalar function $f$ and vector field $\vec v$ we have the multi-dimensional analog of the product rule for derivatives $$\nabla\cdot(f\vec v)=(\nabla f)\cdot\vec v + f\nabla\cdot v$$ ...
Lieven's user avatar
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