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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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1answer
37 views

What is the difference between zero vector and null vector?

What is the difference between zero vector and null vector? I think it is not a duplicate.
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0answers
16 views

Derivative of scalar wrt vector and derivative of scalar wrt matrix

It seems to me that the derivative (or more precisely) the gradient of a scalar valued function with respect to (wrt) a vector can, at least in most contexts be considered equivalent to the gradient ...
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0answers
24 views

Finding the direction using vectors

I have this question that I have no idea were to start and I would like a push in the right direction. A boat has a sail surface of 10m^2 and traveling N40°E. And wind is 6m/s from N30°W. What is the ...
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1answer
14 views

Apply the Hadamard transform to different state vectors

I am in the process of understanding a proof. First, the following is said there: $$H\begin{pmatrix}1\\0\\\vdots\\0\end{pmatrix}=\frac{1}{\sqrt{N}}\begin{pmatrix}1\\1\\1\\1\end{pmatrix}$$ This is ...
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1answer
13 views

Solving plane geometry using vectors and conditions on angles

I can use simple vectors to work out Euclidean properties of plane figures such as the location of the intersection of the bisectors of the sides of triangles (the centroid). Is it possible to use ...
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2answers
13 views

Transformation matrix to align an object towards a vector.

I have an object(say, a prism) in 3D space and a vector. I need to align this object in the direction of the given vector. What would be the transformation matrix required to achieve it? Assume that ...
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0answers
21 views

Find angle of rectangle within rectangle [on hold]

I have been very unsuccessful in deriving the formula to get the angle in the image linked for a rectangle of given width (2) of unknown length within a rectangle of given width (13) and given the ...
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0answers
30 views

Is the derivative of T(t) over the magnitude of T(t) always the unit normal vector?

If you have an equation $$r(t)=\langle 1,\cos(t),\sin(t)\rangle$$ then $$r'(t)=\langle 0,-\sin(t),\cos(t)\rangle $$ and $$\|r'(t)\|=\sqrt{\sin^2(t)+\cos^2(t)}=1$$ thus $T(t)=\langle 0,-\sin(t),\cos(t)\...
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0answers
14 views

velocity gradient in cylindrical coordinate.

In cartesian co-ordinate ($x,y,z$), gradient of velocity($\mathbf{u}=(u_x,u_y,u_z)$)(Jacobian matrix)is defined: \begin{equation*} \nabla \mathbf{u}= \begin{pmatrix} \partial_x u_x && \...
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1answer
19 views

Divergence and curl of vector

A vector field f would be in the form of a homogeneous, plane wave, $$\vec{f}=\vec{f}(\tau)$$ $$ \tau =t-\vec{k}\vec{r}/c$$ where t is the time coordinate, the one-vector k the fixed propagation ...
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1answer
29 views

Factoring out a vector from a matrix product

I have the following expression $$ x^\top + a x^\top B = 0$$ where $a$ and $x$ are column vectors and $B$ is a matrix. Dimensions are such that matrix products work everywhere. Now my question ...
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3answers
33 views

Help finding the equation of a cone in space

The line $\frac{x-2}{2}=\frac{y+1}{-2}=\frac{z+1}{-1}$ Is the axis of a circular cone with vertex on the xy-plane. Find the equation of the cone, if the point $M_1(1,1,-\frac{5}{2})$ is on the ...
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1answer
36 views

Multiplying a vector by a scalar [on hold]

can someone help me answer this question? The question goes like this: ...
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0answers
12 views

Sigmoid Function Involving Vectors When Performing Network Embedding

I'm currently trying to understand how the graph embedding algorithm described in "LINE: Large-scale Information Network Embedding" works. In order to generate an embedding that represents a large ...
1
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1answer
27 views

On vector notation

I've read that, when expressing a quantum mechanical operator into a particular representation, the more-formally correct notation is $\rightarrow$ rather than $=$, e.g. the momentum operator ...
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0answers
8 views

Area of the base of the prism formed by three planes

The three planes $P_1: kx + y+ z=2$,$P_2:x+y-z=3$,$P_3: x+2z=2$ form a triangular prism and area of the normal section (where the normal section of the triangular prism is the plane parallel to the ...
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2answers
37 views

If $a^2 = b^2$ then which values $a$ and $b$ are constrained to be? [on hold]

I've the following subset of $\mathbb{R}^3$: $$ Y= \{(a, b,c)^T | a^2=b^2\} \subset \mathbb{R}^3 $$ How can I embed the condition $a^2=b^2$ into the vector? That is, what can I say about $a$ and $b$?...
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1answer
49 views

Vector - number * vector =?

I can't wrap my head around these vector subtractions. They make 0 sense to me, can anyone help me get this simple step correct in my head? I'm getting nowhere since a hour. Im trying to use the Gram-...
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3answers
31 views

If two vectors are not parallel, can we still compare them i.e. greater than or smaller than?

If vectors $x$ and vector $y$ are a member of $R^n$, and they are not parallel. Can we say anything about $x>y$ or $x<y$? I know that they won't be equal because if $x =$ [$x_1, x_2,..., x_n$] ...
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0answers
14 views

Projection of a vector on a plane defined by two variable vectors

Let $\vec{a},\vec{b},\vec{c}$ be two non-coplanar unit vectors, equally inclined to each other at an angle of $\theta$. Find the projection of $\vec{c}$ on the plane defined by $\vec{a}$ and $\vec{b}$....
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1answer
40 views

Demonstrate how determine when two mobile will be closest (dp.dv) / ||dv||²

I have two mobile, I now their position and velocity. I want to be able to determine in how many time they will be the closest if they still move at the same velocity. In a book I read, they present ...
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1answer
22 views

Prove the diagonals of a parallelpiped bisect each other

I am stuck on how to Prove the diagonals of a parallelpiped bisect each other I have been given the hint to make one of the corners O. If possible I would just like a push in the right direction. ...
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2answers
38 views

State true or false ( I am not sure what i did wrong)

For 𝐮,𝐯 ∈ ℝ𝑛, we have ‖𝐮−𝐯‖≤‖𝐮+𝐯‖. The dot product of two vectors is a vector. For 𝐮,𝐯∈ℝ𝑛, we have ‖𝐮−𝐯‖≤‖𝐮‖+‖𝐯‖. A homogeneous system of linear equations with more equations than ...
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0answers
25 views

I attempted to visualize dot product of complex vectors. What do you advice?

I am an electronics undergraduate student currently learning wavelets. In the book A First Course in Wavelets with Fourier Analysis authors first introduce complex vectors and their dot products. Then ...
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0answers
14 views

Rotation of vector by rotation matrix

Assume the following expression $$ \begin{bmatrix} a_1^* \\ a_2^* \end{bmatrix} = \begin{bmatrix} \cos(45) & - \sin(45) \\ \sin(45) & \cos(45) \end{bmatrix} \begin{bmatrix} a_1 \\ a_2 \end{...
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1answer
11 views

Finding scalar multiples of two variable Vector sum [on hold]

If a=3i+2j,b=3j-2i and c=12i-5j,find numbers p and q such that pa+qb=c.
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1answer
12 views

Misconception related to Resultant of Vectors [on hold]

I have read in many books that the triangle law gives us resulatant of two vectors. $Question$ : How do we knew that the resultant is sum of two vectors not the product of the two vectors?? Clearly ,...
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1answer
26 views

How to calculate a point after rotation given two unit vectors?

I have two unit vectors: before and after rotation. Point (0, 0, 1) is moved to (-0.42, 0.19, 0.88) after rotation. If I had a point of (-0.066, 0.635, -0.184) before rotation, how it would be ...
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0answers
13 views

Checking if lines intersect or skew from directions ratios and a point on each line

Well , this is what's in my book $DR_1(m1,m2,m3)$ with $A(a1,a2,a3)$ $DR_2(n1,n2,n3)$ with $B(b1,b2,b3)$ \begin{vmatrix} m1 & m2 & m3 \\ n1 & n2 & n3 \\ a_1-b_1& a_2-b_2 & ...
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1answer
38 views

Surface integral of hyperboloid using polar coordinates fails?

I am trying to find the surface area of the hyperboloid $x^2 + y^2 − z^2 = 1$ where $0\le z \le 1 $. My book goes ahead making hyperbolic substitutions, however I don't understand why the simple ...
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4answers
36 views

How to find the plane which contains a point and a line

I know that $\Pi$ contains the point $(2,0,5)$ and the line $\frac{x-10}3 = \frac{y-3}2 = \frac{z-7}2$. How would I find the minimum vector connecting the point and the line so I can then work out the ...
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0answers
9 views

vector operation question - finding sail angles [on hold]

A yacht has a sail surface of 10m/s2 . It is travelling N40^oE. The wind is 6 m/s coming from N30^oW. What is the best angle to have the sail at?
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0answers
31 views

The value of $ ( \hat{i} \cos \alpha + \hat{j} \cos \beta + \hat{k} \cos \delta )$ [on hold]

The value of $ ( \hat{i} \cos \alpha + \hat{j} \cos \beta + \hat{k} \cos \delta )$ is $1) \hat{A} $ $2) A\hat{A}$ $3) \hat{A}\cdot\vec{A}$ $4) 1$ First of all I am not clear as to what it is ...
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0answers
14 views

Calculating which plane intersects a number of vectors so that the intersection points or all on the circumference of a circle

I have a plane in $R^3$ that is defined by: $$ (E - O)/h \cdot (x - O) + h = 0 $$ $E$ is a point located on the plane. $O$ is the origin and h is the distance between $E$ and $O$. $x$ is a point ...
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2answers
26 views

Derivation of Family of Circles Meeting at Two Points [duplicate]

We know the formula of "family of circles intersecting at two points" as follows: $$x^2 + y^2 + D_1 x + E_1 y + F_1 + \lambda (x^2 + y^2 + D_2 x + E_2 y + F_2) = 0, \qquad \lambda \setminus \{-1\} \in ...
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0answers
18 views

Scalar and Vector Projections

As a part of my homework, I was asked to draw a diagram for each question and illustrate the projection. In the textbook, it shows the following diagrams (in questions 9A & 10A) that effectively ...
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2answers
30 views

Linear Algebra - R^2 find distance between point and line [closed]

Q = (2, 2), line l with equation [xy]= [-1 2]+ t[1 1] (vector form of line) now find the distance between Q and l. Since Q is not a normal vector then I cannot use proj as the distance. Any help?
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2answers
49 views

What are sets, matrices, …etc?

I am writing a CS research paper where I'm using sets, matrices, and vectors to solve a particular problem. I have two sets, $R$ and $T$, that will be used throughout the entire solution, and a couple ...
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2answers
28 views

Can anyone help solve a linear algebra problem?

Let $u,v$ be vectors such that $||u|| = 2, ||v|| = \sqrt3$ , and $u \cdot v = 1$. Find ||u + v||. So far I calculated $\dfrac{u\cdot v}{||u||*||v||}=\cos(\theta)$ and then with one angle and two ...
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2answers
36 views

What do matrices do to the vectors?

Let’s consider the matrix shearing transformation. Does it change the space of the vector? So any further transformations that we do will add upon that. Or does it change the relation between vector ...
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1answer
21 views

Solving inhomogeneous vector differential equation

I'm trying to solve: $$\dot{y} = By + \begin{pmatrix} -2t \\ 1+3t \end{pmatrix} $$ Where $B = \begin{pmatrix} 3 & 2 & \\ -5 & -3 & \end{pmatrix}$ I'm also giving the information $y(0) =...
2
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1answer
45 views

Three ways to find the normal of a hyperboloid?

In the hyperboloid $x^2 + y^2 − z^2 = 4$, where $z \ge 0$, I have found three ways to get the normal vector, but my problem is they do not seem equivalent. The first is as I have been taught that you ...
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2answers
14 views

Relation between Vectors and Coordinates [closed]

My teacher said that Vector Algebra and Coordinates are same thing but only difference is sense of direction. $Question$: Do we use vectors just for definiting direction or there is any other purpose ...
2
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1answer
42 views

Does matrix transform the $X$-$Y$ space of the vector?

Does matrix transform the $X$-$Y$ space of the vector, so it's not that the output of calculations is the vector with other data in the same basis vector space? Let's consider the matrix: $$ \...
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0answers
28 views

N-Dimensional Sphere intersections embedded in higher dimensional space

Let's say we have some D dimensional Euclidean space. Let me use the term S-Sphere to only indicate spheres that match the dimensionality of the space they reside in, while Circles are spheres with ...
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4answers
29 views

Proof linear dependency for vectors in $3$-D space

How to proof the linear dependency / independency ONLY using vectors (not through matrixes), as I am not familiar with this concept for now. The example is the following: Are the following vectors ...
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1answer
26 views

magnitude of unknown force on wall bracket question

I'm really stuck with part b of this question, even where to start. I think I keep over complicating it. If anyone could help that'd be amazing!
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1answer
18 views

What are the components of the vector field F(x,y,z)=xa? [closed]

a is a constant vector. Not really sure how to get the i,j and k components from this vector field.
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1answer
17 views

Given vectors AB, BC, and BC Simplify the following expression.

Given vectors $\overrightarrow{AB}, \overrightarrow{DC},$ and $ \overrightarrow{BC}$, simplify $\overrightarrow{AB} − \overrightarrow{DC} + \overrightarrow{BC} $ Here's what I have done: $\...
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1answer
14 views

Vectors and applications

Consider the two vectors $\vec{A}$ and $\vec{B}$ . The sum of their vectors ie: $|\vec{A}+\vec{B}|$ , if $|\vec{A}|>|\vec{B}|$ 1) is equal to $|\vec{A}|+|\vec{B}|$ 2)must be less than $|\vec{A}|+...