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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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Elementwise Matrix Derivative

In the context of a neural network, where $l=1,...,N$ denotes a layer, I have a linear mapping: $$ \tilde{x}^{(l)}:= W^{(l)} x^{(l-1)} + b^{(l)}$$ where $x$ a vector with input, $W$ a matrix ...
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5answers
39 views

Prove that $u\cdot v = \frac{1}{4}||u+v||^2 - \frac{1}{4}||u-v||^2 \forall u,v \in \mathbb{R^n}$

I am trying to prove the above statement but I'm not sure if my proof is correct. My proof is as follows, Given $u\cdot v$, we know by the C-E Inequality that $|u \cdot v| \leq ||u|| \ ||v||$ ...
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0answers
8 views

Erroneous vector filtering from image [on hold]

I am working on the motion estimates using two images by python based algorithm. I got the attached vector field (green color). This field includes some erroneous vector. I want to filter those ...
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1answer
14 views

How to calculate the vector intersecting a sphere tangent and plane

I have a sphere centred at a point (x, y, z) = (0, 0, 0) with radius r = 1. I have a point P on the outside of the sphere. How could I calculate a vector at P, which points along both the tangent ...
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1answer
21 views

Given: $u, v \in \mathbb{R}^3$ find all vectors $w$ such that $u \cdot v = w \cdot u$ [on hold]

I have the following problem, but I do not know how to approach it. Given: $u, v \in \mathbb{R}^3$ find all vectors $w$ such that $u \cdot v = w \cdot u$ (dot product) Can anyone give me a hint on ...
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3answers
50 views

Proof that $\| u \|=\| v\|\iff\langle u+v,u-v\rangle=0$

Let $(\cdot,\cdot):\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}^n$ be any map that fulfills the properties $u,v,w\in\mathbb{R}^n;\; \lambda\in\mathbb{R};\; \langle u;v\rangle:=(u_1v_1)+\cdots+(...
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1answer
36 views

Vectors dot product

Could someone please spot the error I've made in this question: Q: A theme park has two zip wires. Sarah models the two zip wires as straight lines $(L_1, L_2)$ using coordinates in metres. The ends ...
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1answer
35 views

How can we justify $\vec F = \frac{d \vec p}{dt}$ by just vector subtraction? [on hold]

I want to justify Newton's second law for the linear momentum of a particle: $$\vec F = \frac{d \vec p}{dt}$$ using really basic linear algebra. Basically, just with vector subtraction. This is ...
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0answers
15 views

Subtraction of vectors in spherical space

I have 4 microphones placed in a spherical coordinate system. I know the $(r_i,\theta_i,\varphi_i)$ for each microphone $m_i$. Given a speed of sound $C$ and the direction from which the sound arrives ...
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0answers
22 views

Write a sequence of vectors as a matrix

Let $X \in \mathbb{R}^{N\times T}$, where $N,T\in \mathbb{Z}^{> 0}$, and $x\in (\mathbb{R}^N)^T$, by which I mean $x = (x_1, x_2, \dots, x_T)$, where $x_i \in \mathbb{R}^N$. The objects $X$ and $x$...
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1answer
26 views

Linear span of given vectors expresses a plane in $\mathbb R^3$?

Given vectors $(1, 3, 5), (-2 , -6, -10)$ and $(2, 6 , 10)$ determine whether the linear span of the above is a plane in $\mathbb R^3$. The vectors are linearly dependent nd hence do not form ...
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0answers
19 views

Signed angle between higher-dimensional oriented vectors?

I am working with vectors in $\mathbb{R}^4$. Any two such (non-parallel) vectors obviously define a plane, and I can rotate any vector in the plane defined by itself and a second vector as follows: $...
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1answer
28 views

How to get a basis of $U \cap V$ where U and V are the column space of $A$ and $B$.?

Problem: Let $A = \begin{bmatrix}5& 2& -1\\3& 1& 0\\ -1& 0& -1 \end{bmatrix}$, $B = \begin{bmatrix} 4& -3\\ -2& 3\\ 1& -2\end{bmatrix}$, $U = C(A)$ and $V = ...
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0answers
15 views

Finding two unknown scalars using vectors GCSE

Asking about question 9d) in attached photo. I have the mark scheme to this question (also attached) and I understand the principle of 9d). I think you have to equate it somehow, using the ...
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1answer
28 views

Linear dependence of three vectors

Could someone please explain why, if two vectors in a set of three are parallel to each other, that this implies that the whole set of three vectors is linearly dependent? I have tried to show this ...
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2answers
48 views

Is a quaternion a way to divide vectors? [on hold]

This is a naive question but I read popular science articles where it is stated that quaternions define vector division, without further explanations
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1answer
50 views

Determining value of product $ (2\vec{a}+\vec{b}).[(\vec{a}\times\vec{b})\times(\vec{a}-2\vec{b})]$ [on hold]

enter image description hereenter image description hereenter image description hereIf $\vec{a}$ and $\vec{b} $ are vectors in space given by $\vec{a} = \frac{\hat{i}-2\hat{j}}{\sqrt{5}}$ and $\vec{b}...
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1answer
37 views

vectors and scalars… [on hold]

If $\vec u=\hat i×(\vec a×\hat i)+\hat j×(\vec a×\hat j)+\hat k×(\vec a×\hat k)$, then: (A) $\vec u$ is a unit vector (B) $\vec u=\vec a+\hat i+\hat j+\hat k$ (C) $\vec u=2\vec a$ (D) $\vec u=...
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0answers
15 views

Understanding the orthogonal projection vector derivation

As you can see below, $z$ is the projection of $x$ onto $y$... I am trying to derive the orthogonal projection formula based on things I already know. Calculating $cos(\theta)$ is trivial... $$cos(\...
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0answers
30 views

How to find the rotation vector by deriving the final vector with respect to the displacement?

My understanding of a rotation of a vector can be done by using a 2D rotation matrix as shown below, $R(\theta )=\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \\\end{...
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2answers
48 views

Show that a vector is equal to the reciprocal of its reciprocal

Given three vectors $\vec u$, $\vec v$, $\vec w$, and their reciprocals $$ \vec u^{\,\prime} := \frac{\vec v \times \vec w}{(\vec u,\vec v, \vec w)} \qquad \vec v^{\,\prime} := \frac{\vec w \times \...
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0answers
37 views

Which of the following are true for the vector $\overrightarrow A$? [on hold]

The vector $\overrightarrow A$ has magnitude $A$ and $\hat A$ is a unit vector in the direction of $\overrightarrow A,$ then which of the following are correct: $1)$ $\overrightarrow A \cdot \hat A= ...
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0answers
6 views

Unit vector Identicon

Is there a way to visualize multiple n-dimensional ($n\approx300$) unit vectors with the property that large changes to the vector result in large visual changes, but small changes result in little to ...
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1answer
39 views

How to find a vector perpendicular to plane ABC given A,B and C

Let $A=(1,0,1)$, $B=(2,1,-1)$, $C(0,1,2)$ Find a vector perpendicular to the plane $ABC$. the solution I was given by my lecturer: Does it matter which vectors I use? Because my attempt has got ...
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1answer
15 views

Directional Derivative in the direction in which $z$ is growing

Find the directional derivative of $f(x, y, z) = xy + 2xz - y^2 + z^2$, at the point $P = (1, -2, 1)$, passing through the curve $x = t, y = t -3, z = t^2$, in the direction in which $z$ is growing. ...
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1answer
27 views

Deriving the scalar/dot product without using the dot product itself?

How do you derive the result a.b = |a||b|cosθ I've seen some geometric derivations using the cosine rule but they then use the definition of a dot product to solve that but I wondered what a full ...
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1answer
52 views

What's the point of the cross product?

I don't understand the motivation behind defining cross products the way they're defined. Given two vectors $\vec{A}$ and $\vec{B}$ in $\mathbb{R^3}$, I can find a third vector $\vec{C}$ such that $\...
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2answers
28 views

Find the position vectors of points B and C, both lying on a line, such that the length AB = AC = 10

The position vector of the point $A$ is $2\vec{i} - \vec{k}$ and the equation of the the line is: $$\vec{r} = (-7, 15, -5) + \lambda (3, -7, 4) \ . $$ Find the position vectors of points $B$...
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2answers
29 views

Finding the angle between a line and a plane

Given that the equation of the line is: $$ \mbox{P:}\quad \left\{\begin{array}{rcrcrcr} 3x & - & y & + & z & = & 6 \\ x & + & 2y & + & z & = &-3 \end{...
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1answer
19 views

How to see if a subspace is contained in another subspace?

So basically I have a problem with 2 subspaces given in the following spans $$U=\mathscr L\{(1,2,-1,3),(2,4,1,-2),(3,6,3,-7)\}$$$$V=\mathscr L\{(1,2,-4,11),(2,4,0,14)\}$$ And I am asked if it is true ...
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0answers
17 views

Distance between a point and a line in space with unknown line equation

We have $A(-2,3,1)$ and we have to find the distance from $A$ to line which contains point $P(-3,5,2)$ and this line makes equal points with coordinate axis. I know how to solve this, I need the ...
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0answers
18 views

Vectorial and parametric equation

I was solving some vectors exercises but I came across with some doubts about them. I don't know how to do this exercise, so I would appreciate some help. Thanks. 1) Find a parametric and a vectorial ...
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1answer
10 views

Given a graph with a position vector and a vector with velocity, defining another vector. No idea where to start.

Diagram for question, click for image Hi Please see image link for the question (because there is a diagram) I have no idea to where even start with this! All help appreciated, thanks
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3answers
53 views

How is the following decomposition done?

\begin{align*} \begin{bmatrix} 1-a & a \\[5pt] b & 1-b \end{bmatrix} &= \frac{1}{a+b} \begin{bmatrix} b & a \\[5pt] b & a \end{bmatrix}+\frac{1-a-b}{...
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0answers
9 views

Vectors and Coordinate planes with non-perpendicular axis

Given two vectors $A$ and $B$ with $|\theta_A-\theta_B|=\frac{\pi}{2}$ and $r_a$ and $r_b$ are any real numbers, can every possible vector be represented by $A+B$ with some constant $r_a$ and $r_b$? ...
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1answer
35 views

How to find distance between a plane and a vector [closed]

I followed the steps from another answer on this site but I arrived to the wrong answer and didn't really understand what was happening, my answer was $\frac{9\sqrt{35}}{32}$ but the two equations ...
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1answer
27 views

Prove using vectors two lines bisect each other

I have the points $O, A, B$ and $C$. Relative to $O$, the position vectors of $A$, $B$ and $C$ are $(1,4, 2 )$, $ (3, 3, 3) $, $( 2, -1, 1)$ Want to show that the lines $OB$ and $AC$ bisect each ...
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0answers
13 views

Hesse normal form $xcos\theta+ysin\theta=ρ$

How should I prove that the angle between the x-axis and the perpendicular line from the origin to the straight line of $$xcos\theta+ysin\theta=ρ$$ (ρ≥0) is the θ. Can someone guide me through this ...
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1answer
35 views

Find the components of a vector?

Can I find the components of a vector while I know it's norm ? I also know it is collinear to another vector of which I know the components. Here is an image :
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0answers
28 views

Vectors and straight lines

I was solving some vectors exercises but I came across with some doubts about them. I don't know how to do these exercises, so I would appreciate some help. Thanks. 1) Find a parametric and a ...
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1answer
14 views

Resolving Cross Product Ambiguity Algebraically

Say we have an orthonormal basis in $\mathbb R^3$ $\{A, B, C\}$. Then when taking the cross product of any 2 of these, we know it is equal to either the third basis element, or $-1$ times that element....
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2answers
198 views

How to write vectors in abbreviated set notation?

I was wondering whether anyone knew how to write a vectors in abbreviated set notation to express the solutions to this question: "Determine all values of x, y, z ∈ R such that (x, y, z) is ...
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1answer
26 views

How do I solve this vector question? [closed]

find vector, parametric and scalar equations of the plane that contains the point $(2,7,-3)$ and the line $\frac{x}{6} = \frac{y}{2} = \frac{z}{5}$
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0answers
18 views

Normal Vector in the place

I've seen two definitions of a normal vector to a curve in $\mathbb{R^2}$. Suppose we have a parametrisation of our curve: $r(t)=(x(t),y(t)),$ Then differentiating once gives us a tangent vector, ...
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0answers
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Finding Cartesian Equation of planes [closed]

Find the Cartesian equation of the plane(s) 2 units from the point $A(0,2,-1)$ and perpendicular to the line of intersection of the planes given by $r = (3,-2,2)+s(-2,1,0)+t(4,-3,2)$ and $3x+3y+4z-4=0$...
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1answer
30 views

Prove that Distributive law holds: $(\lambda+\mu)u=\lambda u+\mu u$ and $\lambda(u+v)=\lambda u+\lambda v$

Let $u,v,w$ n-tuples (vectors) and $\lambda,\mu$ any number. Prove that Distributive law holds: $(\lambda+\mu)u=\lambda u+\mu u$ We defined in our lecture, that Let $u=\left(u_1,u_2,\cdots,u_n\...
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0answers
13 views

Vector Difference

I realize this may be an elementary question however I want to make sure I am not mistaken in my analysis. I am uncertain what property of $\nabla$ means these two terms are not equal. (It came up in ...
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3answers
40 views

If $\|u+tv\| \ge \|u\|$ for all $t$, prove that $u \cdot v=0$

Let $u, v \in \mathbb R^n$. Prove that if $$\|u+tv\| \ge \|u\|$$ for all $t \in \mathbb R$, then $u\cdot v=0$ (vectors $u$ and $v$ are perpendicular). I tried writing $v$ as $(n+xu)$, where $u\cdot n=...
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0answers
16 views

How do I find a point on the surface of a sphere that's within a set range of another point?

I need to find a point in 3d space that's a set distance away from one point and within a set range of another. How do I do this?
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0answers
14 views

Perspective in computer graphics

Looking at some of the math problems (involving vectors) on perspective drawing in computer graphic. But I am a bit confused by the different authors as some placed the object in between the eye and ...