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Questions tagged [vectors]

Use this tag for questions and problems involving vectors, e.g., in an Euclidean plane or space. More abstract questions, might better be tagged vector-spaces, linear-algebra, etc.

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3 views

Coplanar points and vectors pointing them from 0xyz axis

Let 4 points be A,B,C,D and vectors starting from(0,0,0) in xyz axis a,b,c,d accordingly(for example the vector 0A is a) If the 4 points A,B,C,D are coplanar then proof [(a,b,c) is the scalar triple ...
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1answer
17 views

Linear algebra identity evaluation

I really couldn't find anything related to this simple identity I came up with so: $$\vec{r}=(r_x,r_y)=(r_x, \angle0)+(r_y,\angle\frac{\pi}{2})$$ My thinking process was that $r_y$ is practically ...
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0answers
2 views

what is the result type 3D vector, point and scalar arithmetic operations?

I doing programming but I am not good at math. I want to represent point and vector in different type. Both point and vector are tuple of 3 elements, where point represent location and vector ...
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1answer
17 views

What is the vector form of Taylor series expansion?

What is the expression for expansion of $\phi(\vec r+ \vec l)$ where $\vec r$ is variable and $\vec l$ is a constant vector. I think it can be expanded as a vector form of taylor series as $\phi(\vec ...
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1answer
20 views

Triple scalar product

So I came across this question: Given vector $\textbf{u} = i+j, \textbf{v} = j+k, \textbf{w} = i+k$. Find the triple scalar product $u(\textbf{v}\times \textbf{w})$. So I tried to check my notes to ...
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2answers
21 views

Show that $a,b,c$ forms the sides of a triangle. Please help on my attempt.

Show that $a=2i+2j+3k,b=3i+j-k,c=i-j-4k$ forms the sides of a triangle. My attempt: $|a|=\sqrt{17},|b|=\sqrt{11},|c|=\sqrt{18}.$ Since $|c|<|a|+|b|$ using triangle inequality, we can say $a,b,c$ ...
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0answers
8 views

rotate 3d unit vector a, on plane of a and the j (up) axis

vector $\tilde a = c\tilde i+d\tilde j+e\tilde k$ is our input, and vector $\tilde b = f\tilde i+g\tilde j+h\tilde k$ is our output, and $\theta$ is the angle to rotate by. Essentially $\tilde b = f(\...
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1answer
14 views

Find a function M

Let $x(t)$ be a real valued vector. Can you find a function M such that $\dot{M}=\frac{\text{d}M}{\text{d}t}=\dot{x}^T\dot{x}$. I have tried $M=\dot{x}^Tx, M=x^Tx$ and many more which don't work. ...
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0answers
9 views

Arrangements of terms in the dot and cross products [on hold]

Why is the arrangement of terms in the cross product important and not important in the dot product
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1answer
40 views

Matrices: help with homework

I need to prove that $\vec{x}$ is a solution of $A\vec{x}=\vec{b}$: $$ \begin{vmatrix} 2&-7&-3\\ -4&1&5\\ 1&3&-1\\ \end{vmatrix} \cdot \begin{vmatrix} 5\\ -1\\ 7\\ \end{...
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0answers
16 views

Check if 3D Point is below a line

Is it possible to know if a point in below a direction vector or a line? I'm not good at maths and I'm having a really hard time figuring this out, I seen some answers to doing this in a 2D but can't ...
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1answer
57 views

How to fill “holes” in a vector ? [1 0 1] -> [1 1 1] [on hold]

Support I have a vector that looks like this: $$[\,0\ 0\ 1\ 0\ 1\ 0\ 0\ 1\ 1] $$ What algebraic operation will fill the "holes" between $2$ "$1$"s and makes it a squence of "$1$"? What I mean is ...
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2answers
39 views

What's the cross product in 2 dimensions? [duplicate]

The math book i'm using states that the cross product for two vectors is defined over $R^3$: $$u = (a,b,c)$$ $$v = (d,e,f)$$ is: $$u \times v = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\...
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1answer
19 views

Average direction between two vectors

this is my first time asking a question so I'm sorry in advance for any mistake I might make. So I have 3 points in 3D space: A, B and C. What I want to do is have an object on point B point towards ...
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0answers
42 views

Serge Lang (introduction to linear algebra)

I am reading Introduction to linear algebra by Serge Lang. I have a confusion. I read here on website that we cannot add points. But he defines this operation. how can this be?
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0answers
2 views

Orientation(arrangement) of the terms of a cross product or dot product

when taking the cross product of 2 vectors, why is their arrangement important, and not important when taking their dot product? (why is Ā×Ū=-Ū×Ā in the cross product, but Ā×Ū= Ā×Ū in the dot product)
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0answers
27 views

Introduction to linear algebra (Strang) 1.1 problem 20

Under what restrictions on c, d, e, will the combinations c$u$+d$v$+e$w$ fill the dashed triangle? ($u$, $v$, $w$ are 3-d vectors) I have been trying to see the way I could make restrictions on c, d ...
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0answers
21 views

vectors equation of a plane [on hold]

Does the line with equation (x, y, z) = (5, -4, 6) + u(1,4,-1) lie in the plane with equation (x, y, z) = (3, 0, 2) + s(1,1,-1) + t(2, -1, 1)? Justify your answer algebraically.
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2answers
25 views

Vector Equation of a plane

The lines $L1$ and $L2$ have equations $r = 8i - 14j + 13k + s (-4i + 7j - 6k)$ and $x/2 = (y-17)/5 = (z+7)/-1$ respectively. The plane contains both $L1$ and $L2$: a) Find the vector equation of the ...
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0answers
17 views

Vectors and line equations and planes

Which of the three coordinate planes does the line given by $x=16t, y=-4-9t, z=34$ intersect? I am confused where to start with this question. I was thinking maybe to solve $t$ from equation of $y$ ...
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0answers
29 views

Mistake in calculations related to vectors and norms

I am rather new to calculus, and am trying to resolve the following question. I have come to an answer, but it is not listed amongst the possible answers, so I would love to know where my reasoning ...
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0answers
43 views

Real valued vector representation of Hermitian matrix

Since my application is in physics, I created a specialized version of this question on physics.SE. One can write a symmetric matrix $M \in \{\mathbb{R}, \mathbb{C}\}^{n \times n}$ in a half-...
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0answers
32 views

Linear Algebra question. Finding vector component of $\bf u$ orthogonal to $\bf a$ [on hold]

Let $\mathbf u = (2, 1, -1),\mathbf a = (-3, 2, -1)$. Find the vector component of $\bf u$ orthogonal to $\bf a$. I followed a solution I found on the internet but the system keeps saying half of the ...
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1answer
18 views

Degree between vector and point

I have a vector and a point $(x, y)$. The vector starts from $(0, 0)$ and goes to $(x_1, y_1)$. $x$, $y$, $x_1$, $y_1$ are known. How can I get the degree that vector should rotate clockwise to face ...
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1answer
17 views

Find the vector projection and component perpendicular

Question: If vector $a = <2,3,5>$ and vector $b = <2,-2,-1>$, find the vector projection of b onto a and hence find the component of b perpendicular to a. I found the vector projection of ...
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6answers
500 views

Proving $(\bf x\times y\cdot N)\ z+(y\times z\cdot N)\ x+(z\times x \cdot N)\ y= 0$ when $\bf x,y,z$ are coplanar and $\bf N$ is a unit normal vector

Prove that if $\mathbf{x},\mathbf{y},\mathbf{z} \in \mathbb{R}^3$ are coplanar vectors and $\mathbf{N}$ is a unit normal vector to the plane then $$(\mathbf{x}\times\mathbf{y} \cdot \mathbf{N})\ \...
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0answers
27 views

Where can I find a broad set of exercises on Matrix calculus? [duplicate]

I am looking for exercises particularly on matrix differentiation - any reference textbook with theory examples is appreciated too.
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0answers
21 views

Gram-Schmidt orthogonalization process with specific dot product

I have three vectors $$v_1=(1,1,1)^T$$ $$v_2=(1,1,0)^T$$ $$v_3=(1,0,0)^T$$ and special dot product definition $$(\overline{(x_1,x_2,x_3)},\overline{(y_1,y_2,y_3)})=2x_1y_2+x_1y_1+2x_2y_1+x_3y_3$$ I ...
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1answer
31 views

Dot product for orthonormal basis

I want ask which must be the dot product for vectors (1,1,0), (1,0,1), (0,1,1), so they can form a orthonormalbasis.
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1answer
30 views

Basic application of “velocity”

I'm a basic learner. I have been taught: If anything is moving around in a circular path, it's velocity is zero. I want to know that what the velocity really is, what its purposes are. Tell me ...
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1answer
21 views

Getting different answers for same problem on divergence and curl.

Given that $\vec{a}$ is a constant vector and $\vec{r}$ is a position vector. We are asked to prove the following: $$\nabla\times(\vec{a}\times\vec{r})=2\vec{a}$$ I tried two ways. Could prove it ...
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1answer
39 views

How to put an arrow above a letter? [closed]

Can anyone help me put $\rightharpoonup$ above $C_3$ in latex? Need to do so to indicate line integral for Green's Theorem. Any help would be appreciated.
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0answers
20 views

Finding iterative relationship between vectors, involving substitution of a product

Consider $A_N\in\mathbb{R}^{n\times n}$, $V=(\mathbf{v}_1,\mathbf{v}_2,\dots, \mathbf{v}_N)^T$, $\mathbf{v}_i, \mathbf{u}\in\mathbb{R}^n$, is it possible to express, $$\mathbf{v}_{N+1}=\frac{A_N^{-1}...
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0answers
14 views

Jacobian of a skalar function with multi-dimentional vector input

I am trying to compute the Jacobian of $f : \mathbb{R}^{8} \rightarrow \mathbb{R}$, where: $f(\vec{x})= g(T(\vec{x}))= g(\vec{\mathbf{c}})=\Biggl| \|\mathbf{V}\|_{2}^{2} - \|\mathbf{A} \cdot c\|_{2}...
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1answer
40 views

Pythagoras theorem in oblique coordinates

consider two oblique oblique basis vectors of unit length $\vec{r_1}, \vec{r_2}$ then any vector $\vec{v} = p\vec{r_1}+q\vec{r_2}$ define the dot product between two vectors a and b as $|b|$ (ie ...
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0answers
13 views

Show that a point is closest to another point with extreme value theorem

Show there is a point of the plane $\{x \in \mathbb{R^3} \mid x_1 + 2x_2 + 3x_3 = 13\}$ closest to the point $(1, 1, 1)$. Let a function $f: A \rightarrow \mathbb{R}$ defined for all $x \in A$ by $f(...
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2answers
22 views

Prove $(X \theta - \vec{y})^T (X \theta - \vec{y}) = \theta^T X^T X \theta - \theta^T X^T \vec{y} - \vec{y}^T X \theta + \vec{y}^T \vec{y}$

I'm studying Machine Learning Stanford's CS229 course and in the lecture note, page number 11, I'm not getting how does step 2 arrive from step 1 above? Prof. Andrew Ng says that it is the expansion ...
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2answers
27 views

What is the 'meaning' behind $r·n = a·n$

I was confused about what the dot product represents and I think I have grasped that now from this post What does the dot product of two vectors represent? . However, I still cannot understand what $...
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2answers
25 views

Finding an equation of a plane through the origin that is parallel to a given plane and parallel to a line.

A plane through the origin is perpendicular to the plane $2x-y-z=5$ and parallel to the line joining the points $(1,2,3)$ and $(4,-1,2)$. Find the equation of the plane. Analyzing this problem I ...
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2answers
32 views

Proof of Cauchy-Schwartz inequality with dot product and euclidean norm

I have some problems on understanding the proof of Cauchy-Schwartz inequality from my textbook: Given $\textbf{x,y} \in \mathbb{R} \Rightarrow \vert \textbf{x}^T \textbf{y} \vert \le \Vert \textbf{...
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1answer
38 views

$\mathbb{E}[\operatorname{sign}\langle v,z\rangle]$ for $v$ fixed, $z_i\sim N(0,1)$

I am trying to evaluate $\mathbb{E}[\operatorname{sign}\langle v,z\rangle]$ for $v\in\mathbb{R}^n$ fixed and $z_i\sim N(0,1)\ \forall\ i\in[n]$. The $\operatorname{sign}$ part is what is confusing ...
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1answer
46 views

Is this possible in vector calculus?

$$(\boldsymbol{\nabla}\alpha)\wedge(\boldsymbol{\nabla} \wedge \boldsymbol{x} )$$ In all the examples in lecture, it has always been a $$\boldsymbol{\nabla}$$ on the left hand side. Does this give ...
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0answers
28 views

How to fiind the relationship between a given plane equation and a line equation?

Given some plane equation P $x-3y+2y = 4$, and line l $r(t) = <4t, 2t, 2+t>$, how do I find their relationship? For example, are they orthogonal? Parallel? Is l contained in P? I would ...
4
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2answers
53 views

Question about vectors and geometry in calculus

I am rather new to calculus, and was asked to answer the following question. I think I have the right answer, but I would love some feedback, particularly because some answers I derived empirically, ...
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0answers
19 views

Velocities calculation in X Y Z directions for GNSS from Speed [closed]

The Speed and location coordinates at each GNSS location are given. How can be the velocities along X, Y & Z directions at each individual GNSS location calculated ?
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0answers
24 views

Understand reflections in the plane

A plane is spanned by the vectors $\vec{a}$ and $\vec{b}$. The angle between the mirror axes $\vec{a^\perp} $ and $\vec{b^\perp}$ is equal to the angle of the two vectors $\vec{a}$ and $\vec{b}$. Let'...
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1answer
30 views

Does the line passing through $(3,4,-1)$ which is normal to $x+4y-z = -2$, intersect any of the coordinate axes? [closed]

Does the line passing through $(3,4,-1)$ which is normal to $x+4y-z = -2$, intersect any of the coordinate axes? I'm not sure how to go about this question. Any help would be greatly appreciated. ...
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3answers
27 views

What is the meaning of angle between two planes?

Suppose there are two planes that have normal vectors n1 and n2. Let vector n2' = -n2. So, n2' is also the normal vector of plane 2. We can see that the angle between n1 and n2, and between n1 and n2' ...
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votes
1answer
23 views

How to show this vector field is irrotational?

I have the field: $$\bar a(\bar r)=r \bar c + \frac{(\bar c\cdot \bar r)}{r}\bar r$$ where $$\bar c $$ is a constant vector. I have worked through the problem and I cant seem to easily show that: $$ \...
0
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0answers
22 views

Matrix/Vector Addition

If the sum of vectors v and w as shown in the image above is (5,1) and the difference is (1,5). Can we fill the entire xy plane usimg combinations of v and w?