3
votes
1answer
23 views

Why should we expect the divergence operator to be invariant under transformations?

A lot of the time with vector calculus identities, something that seems magical at first ends up having a nice and unique proof. For the divergence operator, one can prove that it's invariant under a ...
0
votes
0answers
3 views

Gentle introduction to discrete vector field [closed]

I am looking for a gentle introduction to discrete vector field. Thanks in advance.
0
votes
1answer
26 views

Vectors and Planes

Let there be 2 planes: $x-y+z=2, 2x-y-z=1$ Find the equation of the line of the intersection of the two planes, as well as that of another plane which goes through that line. Attempt to solve: the ...
0
votes
0answers
25 views

Vector Question Help

A plane is determined by $(x,y,z) = (1,-1,0) + t(1,-1,2)$ and point $p(1,2,3)$. find point of intersection of $(1,4,-1)+s(-6,2,-4)$ with this plane. I tried this: given the data plane equation is:$$ ...
0
votes
1answer
21 views

Question about vector equations of lines and planes

Find the equation of the line going through the point $(2,-3,4)$ ,and which is perpendicular to the plane $ x+2y + 2z = 13$ So I tried this: the normal of the plane is $(1,2,2)$, random point on the ...
0
votes
2answers
44 views

Vectors Question

I have a question regarding Vectors; Find the equation of the plane perpendicular to the vector $\vec{n}\space=(2,3,6)$ and which goes through the point $ A(1,5,3)$. (A cartesian and parametric ...
0
votes
0answers
17 views

How to find a resultant vector for given multiple vectors lie on different position.

I have modelled rectangular features and then I compared my model with a reference model. (Assume everything is in 3D space) . In the below figure, Reference model is shown in light pink ...
0
votes
0answers
26 views

Books which explain vector analysis/algebra in detail.

I'm trying to learn vectors but I can't find a decent book which explains vectors in depth. I need a book which explains vectors from the beginning, using a beginner's approach(assuming the reader ...
1
vote
1answer
51 views

Which of these things is not like the others?

What's in a name? Well quite a lot, if you're confused enough. I have an engineering-style mathematics education, based on good old hand waving and learning bits and pieces from all over the place. I ...
0
votes
0answers
9 views

Cross Product of Covectors

Is the vector/cross product defined for covectors (in the dual space) or is it, strictly speaking, only defined for vectors themselves? I would imagine that it works fine for covectors but I wanted to ...
0
votes
1answer
31 views

How to prove sum of vectors with same magnitude is equal to zero.

Suppose that we have $n$ vectors $v_1,v_2...v_n$ with same magnitude in plane s.t. the angle between $v_i$ and $v_{i+1}$ is $2\pi/n$ then $v_1+v_2+...v_n=0$ for all $n \geq 2$. I can show this by ...
0
votes
3answers
60 views

Explain Normalization in Layman's term

Can someone explain me what is Normalization in Layman's term ? If we have a vector a, we normalize it by dividing it by |a|. That is $$\frac {a}{|a|} $$ Why we need normalization?
3
votes
3answers
70 views

Vector multiplication. Difference between scaler and dot product?

We just started a new class where the first topic is briefly talking about vectors and vector multiplication. All tying this into neural networks. I am a bit behind with the understanding of what the ...
0
votes
2answers
45 views

Vectors Calculation Question

When two vectors are sketched from a single point, the angle between them is θ. Show that the size of their vector summation is given in the expression: $ \sqrt{A^2 + B^2 +2ABcosθ} $. Any ...
4
votes
2answers
64 views

What have Vectors and Matrices got to do with each other?

In my undergraduate course work I learnt Vectors (as in those in vector space with magnitude and direction) separately from Matrices - an $n \times m$ array of numbers. However, after sitting in for a ...
1
vote
0answers
28 views

sweeping edges till they get a given elevation on an oblique plane

I am constructing wireframe model of 3d objects (prisms,..etc.). from a triangular mesh, I have obtained boundary points and fit striaght lines in order to get polygon edges refering to prism ...
1
vote
1answer
58 views

$X_0\subset X$ finite codimension. Show there exists subspace $Z$ s.t. $X= Z\oplus X_0$

I'm not so familiar with these types of arguments concerning vector spaces, and direct sums. Let $X$ be a vector space and $X_0\subset X$ with finite codimension, i.e. $dim(X/X_0)$ is finite. Show ...
0
votes
3answers
65 views

algebraic definition of vector product

I am struggling to justify the the consistency of algebraic definition of vector products. Say given Two vector A , B A.B = |A|. |B| cos(0) where Lets assume ...
1
vote
1answer
167 views

Need to determine if a subset is a vector space in $\mathbb R^2$

Its been about a year since I have done anything with vector spaces and subsets of vector spaces, and now I've found that I have forgotten a lot of material. I am given this: Suppose $$\mathbf x ...
0
votes
1answer
63 views

Extrema of a vector norm under two inner-product constraints.

If $\langle\vec{A},\vec{V}\rangle=1\; ,\; \langle\vec{B},\vec{V}\rangle=c$, then: \begin{align} max\left \| \vec{V} \right \|_{1}=?\;\;\;min\left \| \vec{V} \right \|_{1}=? \end{align} Consider the ...
-1
votes
2answers
129 views

How to rotate two vectors (2d), where their angle is larger than 180.

The rotation matrix $$\begin{bmatrix} \cos\theta & -\sin \theta\\ \sin\theta & \cos\theta \end{bmatrix}$$ cannot process the case that the angle between two vectors is larger than $180$ ...
0
votes
1answer
53 views

Is this statement about vectors true?

If vectors $A$ and $B$ are parallel, then, $|A-B| = |A| - |B|$ Is the above statement true?
1
vote
1answer
822 views

Collinearity of three points of vectors

Show that the three vectors $$A\_ = 2i + j - 3k , B\_ = i - 4k , C\_ = 4i + 3j -k$$ are linearly dependent. Determine a relation between them and hence show that the terminal points are collinear. ...
2
votes
2answers
92 views

Convex cone question.

Hoi, let $V$ be finite dimensional real vector space with inner product $\left\langle . \right\rangle$ and let $\Gamma \neq \{0\}$ be a closed convex cone. Let $$\Gamma_0^{\perp}:=\{v\in ...
1
vote
5answers
99 views

Can I treat vectors in $\Bbb R^3$ with a component equal to zero as vectors in $\Bbb R^2$?

If I have a vector $(0,4,4)$ and was finding perpendicular unit vectors to this, is it the same case as finding perpendicular unit vectors for the vector $(4,4)$? Meaning a maximum of two possible ...
1
vote
0answers
23 views

vector analysis and reduce to a general expression in theta's

$\vec{A},\vec{B},\vec{C}$ and $\vec{D}$ are unit vectors ($|A|=1,|B|=1,|C|=1$ and $|D|=1$). The angle between the vectors, 1) $\vec{A}$ and $\vec{B}$ is $\theta_{1}$ ...
4
votes
3answers
280 views

Shortest length that a vector can have

I came to the following question from a past exam: The vector $v = (k, k, 3 − k)$ depends on a variable $k$. What is the shortest length of the vector $v$ can have? I know that the answer is ...
1
vote
1answer
70 views

Finding distances/similarity between vectors

Let's say for example I have these three vectors representing these three fields: $$\begin{array}{c|c|c} \text{Cars Sold} & \text{Cars in Lot} & \text{Profit Generated}\\ \hline \\140 & ...
4
votes
2answers
135 views

Vectors transformation

Give a necessary and sufficient condition ("if and only if") for when three vectors $a, b, c, \in \mathbb{R^2}$ can be transformed to unit length vectors by a single affine transformation. This is ...
2
votes
1answer
56 views

To show the inequality $\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$

Let $A\in$ $\mathbb{C}^{p\times q}$ with column $u_1,\ldots,u_q$ and rows $\vec{v_1},\ldots,\vec{v_p}$. show that $$\|A\|\geq\max\{\|u_1\|,\ldots,\|u_q\|,\|\vec{v_1}\|,\ldots,\|\vec{v_q}\|\}$$ and ...
5
votes
3answers
160 views

Find the necessary and sufficient conditions on $A$ such that $\|T(\vec{x})\|=|\det A|\cdot\|\vec{x}\|$ for all $\vec{x}$.

Consider the mapping $T:\mathbb{R}^n\mapsto\mathbb{R}^n$ defined by $T(\vec{x})=A\vec{x}$ where $A$ is a $n\times n$ matrix. Find the necessary and sufficient conditions on $A$ such that ...
5
votes
1answer
58 views

Dot products of three or more vectors

Can't we construct a mapping from $V^3(R^1)$ to $R$ such that $a.b.c = a_{x}b_{x}c_{x}+a_{y}b_{y}c_{y}+a_{z}b_{z}c_{z}$ (a,b,c are vectors in $V^3(R^1)$ ) and more generally $a^n$ , $a.b.c.d.e...$ ...
-2
votes
1answer
365 views

Find if possible an orthogonal unit vector at: 2i + 3j - k and - 2i - 3j + 4k

The question is: Find, if possible, an orthogonal unit vector at: $2i + 3j - k$ and $-2i - 3j + 4k$. $$\left|\begin{matrix} i & j & k \\ 2 & 3 & -1 \\ -2 & -3 & 4 ...
1
vote
2answers
38 views

How are vectors defined in terms of sequences?

I'm reading this Wikipedia article on sequences Sequence where it mentions 'Sequences over a field may also be viewed as vectors in a vector space.' under vectors section. I'm not able to grasp this, ...
1
vote
1answer
54 views

what is a vector of polyhedron?

What does v mean in the following, does it a point inside the polyhedron?
3
votes
1answer
136 views

What is the name of this equation?

I have found this picture but I don't know the name of the equation in it. Another thing, what kind of plots are those in the picture? I have also tried to re copy it: $$ ...
0
votes
0answers
50 views

Knowing $\alpha$ and $\beta$, compute $\gamma$. $vers(\vec v)=(\cos\alpha,\cos\beta,\cos\gamma)$

Knowing that: $\vec v=|\vec v| vers(\vec v)$ $\cos\alpha=\frac{\vec v \cdot \vec i}{|\vec v|\cdot|\vec i|}$ $\cos\beta=\frac{\vec v \cdot \vec j}{|\vec v|\cdot|\vec j|}$ $\cos\gamma=\frac{\vec v ...
0
votes
1answer
79 views

inequality between entries of the vector and $l_2$ norm of the vector

Let $a=(a_1, \ldots, a_n)$ be a vector in $R^n$. I am wondering for which vectors the following would be true: $$ \|a\|_2^2\geq c \sum_{i\ne j,i,j}a_ia_j, \quad i,j=1, \ldots, n. $$ Here $c>0$ is ...
0
votes
4answers
142 views

How to get a new point of a vector when rotated.

I want to obtain the new point of a vector that I rotate like this. When I rotate them, I have the angle of rotation. I want to know x and y, it rotates taking the reference point of 0,0 Thanks
0
votes
1answer
26 views

Determine $\alpha$ for which this vector equation takes place.

Assume $ABCD$ is a parallelogram. $O$ is the intersection of the diagonals and $M$ an arbitrary point in the same plan. Determine $\alpha$ for which the following relation takes place: ...
0
votes
0answers
63 views

Finding point distribution by eigen vectors

First of all I want to tell that my mathematics is poor, so I can’t use correct terms. Sorry for that. I have a point data set. This data represents some cylindrical objects surfaces (not exactly ...
1
vote
1answer
29 views

For which points ($P$) on the $x$-axis, $\angle APB= 90^\circ $?

Let $A =(-2, 3, -2)$ and $B =(-6, -1, 1)$. For which points ($P$) on the $x$-axis, $\angle APB= 90^\circ $? I figured, since $P$ is supposed to be on the $x$-axis, the $y$ and $z$ coordinates ...
0
votes
2answers
194 views

Vector Analysis & Linear Algebra

I'm given a positive number, a unit vector $u \in \mathbb{R} ^n $ and a sequence of vectors $ \{ b_k \} _{ k \geq 1} $ such that $|b_k - ku| \leq d $ for every $ k=1,2,...$. This obviously implies $ ...
1
vote
1answer
111 views

basic vector being hermitian

If the space has a mixed metric signature, not all the basis vectors are Hermitian. Nevertheless, they are defined to be self-adjoint under reversion. The vector transpose conjugate is, ...
8
votes
2answers
5k views

What is the proof that covariance matrices are always semi-definite?

Suppose that we have two different discreet signal vectors of $N^{th}$ dimension, namely $\textbf{x}[i]$ and $\textbf{y}[i]$, each one having a total of $M$ set of samples/vectors. $\textbf{x}[m] = ...
3
votes
0answers
638 views

Three-dimensional vectors and force systems

Full disclosure: this is a homework problem. However, I find myself stuck in the middle. The problem is below As shown, a system of cables suspends a crate weighing W = 350 . (Part C 1 figure) ...
0
votes
1answer
90 views

Bases and inner products

I am not quite sure what this question is asking for: Given $f(\vec{x})=x^2+xy+y^2+yz+z^2+xz$, find a basis for the corresponding inner product on $\mathbb R^3$. (I was told that there is an ...
0
votes
0answers
72 views

clever solution to decomposition of linear products?

There may be a better name for this class of problem, and if so feel free to edit! Imagine a matrix consisting of the following columns: daily return, $\alpha_t$, $factor^1_t$, $factor^2_t$, ... and ...
0
votes
1answer
5k views

how to calculate the angle in the x-y, y-z, x-z plane given only 3D vector direction and magnitude?

Please help me solve this. I have been thinking of all sorts of ways to solve this but can't figure out how :(. Ok here's the problem: I am given a three dimensional velocity vector (i know the ...
2
votes
2answers
2k views

What are the rules for complex-component vectors and why?

I want to take the inverse of a dot product, where both vectors have complex components. In other words, if $\textbf{A} \cdot \textbf{B} = d$, and I know $\textbf{A}$ and $d$, I want to find a ...