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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
35 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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2answers
20 views

Intersection of the two planes

I need help with my vector's assignment!!! Let L be the line of intersection of the two planes x+y+z-1=0 and 2x+3y-z+2=0. Find the scalar equation of the plane that contains the line L and passes ...
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2answers
29 views

Intersection of a plane

I need help with my grade 12 Vector's homework. Can a plane be perpendicular to the x-axis and contain the line $x=z, y=0$? Explain. I really hope someone can answer this question
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2answers
356 views

Find the equation of the plane that passes through the line of intersection of two planes and a point

Find the equation of the plane that passes through the line of intersection of the planes $2x-3y-z +1 =0$ and $3x+5y-4z+2=0$, and that also passes through the point $(3,-1,2)$ $\vec n_1 = [2,-3,-1]$ ...
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2answers
367 views

collision point of circle and line

I'm trying to figure out the collision point of the circle and a line, ultimately it should work in 3D but for now just in 2D to simplify the problem as much as possible. I've created 2 examples here ...
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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
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why any vector can be wriiten as the sum of two components in the row space and nullspace?

My textbook says that: there is a $m\times n$ matrix A, any vector x in $R^n$ can be written as the sum of a component $x_r$, in the row space, and a component $x_n$ in the nullspace: $$x=x_r+x_n$$ ...
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3answers
49 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
184 views

Curl of gradient is not zero

I have heard that for some functions $T$, if we calculate $\nabla \times (\nabla T )$ in $2$-dimensional polar coordinates, then we get the delta function. Why do we get that result?
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3answers
1k views

Find a basis of $A = (\{1, \sin(x), (\cos)^2(x), (\sin)^2(x)\})$ and determining its dimension.

We consider a space F(R, R) of functions of R in R. Let A = ({1, $\sin(x)$, $\cos^2(x)$, $\sin^2(x)$}) Find a basis of the vector subspace of F(R,R) and determine its dimension. So I used the ...
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4answers
61 views

How to show $T$ is bijective in the following condition?

Let $T:\ V\rightarrow W$ be a linear transformation, if $\dim(V)=\dim(W)$, $\{v_1,...,v_n\}$ is a basis for $V$ and $\{w_1,...,w_n\}$ is a basis for $W$. Let $T:V\rightarrow W$ be a linear ...
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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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1answer
674 views

Trace of vectors

Does that sound about right? Given that x is $m\times 1$ and y is $m\times 1$ vectors, show that $ tr(\mathbf{xy'})=\mathbf{x'y}$. Attempt: By using the property of $tr(\mathbf{A'})=tr(\mathbf{A})$,...
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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
23 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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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
2k views

Area of parallelogram 3D vectors

I'm given the following parametrization of a parallelogram $(x,y,z)=(1,1,1)+s\cdot(2,-1,-1)+t\cdot(-1,3,2),s\in[0,1],t\in[0,1]$ I'm now asked to determine the area of this parallelogram as well the ...
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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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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
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
274 views

Why is the dot product the projection of one vector onto another? [closed]

Could you please give me an intuitive explanation why the dot product is defined this way?
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1answer
2k views

If $B$ is added to $A$, under what condition does the resultant vector have a magnitude equal to $A+B$?

If $B$ is added to $A$, under what condition does the resultant vector have a magnitude equal to $A+B$? Under what conditions is the resultant vector equal to zero? My Attempt: Let $\theta $ be the ...
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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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1answer
51 views

How to represent a vector in terms of another vector?

If $u, v$ be two unit vectors. $\textbf{Then how to represent $u$ in terms of $v$? }$ I can find a matrix, $R$ such that $u=Rv$, from trial and error. How to derive the matrix analytically? Unit ...
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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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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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2answers
481 views

Point within a spherical triangle given areas

Consider a spherical triangle like this: where $A_1, A_2, A_3,$ and $P$ are points on the sphere and $t_1, t_2, t_3$ are the proportion of the area of the large triangle contained within the small ...
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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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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
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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2answers
1k views

Finding the intersection of two planes, given their normal vectors

Two vectors $v_1=(-1,1,0), v_2=(1,0,1)$ span a plane $P$ in $\Bbb R^3$. Two vectors $w_1=(0,1,0), w_2=(1,1,0)$ span a plane $Q$ in $\Bbb R^3$. I am asked to show that $P$ and $Q$ are different and ...
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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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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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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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1answer
51 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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0answers
21 views

Baseball Projectile Motion

A baseball batter can hit the ball at an angle between $45°$ and $60°$. At all angles, the initial speed is $30$m/s. There are two catchers, catcher $A$ can run at $5$m/s and catcher $B$ can run at $2....
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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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2answers
579 views

Reciprocal Vector Triple Product

I have got to show: $ [A,B,C] = 1/[a,b,c] $ Where $[n,m,k]$ denotes the scalar triple vector product and $A,B,C$ are reciprocal vectors to $a,b,c$ (non-coplanar, but not necessarily orthonormal). ...
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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 ...