Use this tag for questions involving vectors, which are elements of vector fields.

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flux of a vector field on the surface of a sphere

enter image description here I attached a picture of the question, but basically have to find the flux of a field on the surface of a sphere. Ive tried the divergence theorem but it doesnt seem to be ...
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0answers
5 views

Dipole-Coupling Tensor: Electrostatic Dipole Moments

I've been struggling with this problem today. Here's an image of the question I'm attempting to answer. I'm relatively new to tensor algebra (I've been studying it for about a week or two), and I've ...
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2answers
44 views

How do I rotate a vector 90 degrees in a random direction?

I'm building a tree generator and I'm at the point where I want to have sub branches branch off at right angles off the current branch, in random directions. I have a 3D vector defining the direction ...
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0answers
23 views

velocity and acceleration of a disk rotating at a constant speed

A disk of radius 1 is rotating in the counterclockwise direction at a constant angular speed ω. A bug starts at the center of the disk and moves directly toward edge. The position of the bug at time ...
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1answer
20 views

Integrate this Gaussian in $\mathbb{R}^N$

I'm trying to compute this integral: $$\int_{\mathbb{R}^N} \exp\Big((\vec{x} - \vec{\mu})^T(\lambda \text{A}^T \text{A}+\delta \text{L})(\vec{x} - \vec{\mu}) + U(\delta)\Big)d\vec{x}$$ Where $$\mu = ...
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1answer
21 views

Vector cross product proof vs scalar [on hold]

In the set of $\mathbb{R}^3$ vectors $u,w,v$ does $u \times (v \times w) = (u\cdot w)v-(u\cdot v)w$ ? Note the difference between $\times$ and dot. Hint cross product.
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1answer
19 views

Using line integrals and Green formula to calculate force?

A force field $F$ = -y$^2$I$+x$j acts on a particle which moves on a closed loop formed by the sides of a triangle with vertices at $(0,0)$, $(1,0)$ and $(0,2)$ in the anticlockwise direction. Find ...
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2answers
33 views

Calculate the line integral of a half circle as a standing unit circle?

Calculate the line integral $$ \rm I=\int_{C}\mathbf{v}\cdot d\mathbf{r} \tag{01} $$ where $$ \mathbf{v}\left(x,y\right)=y\mathbf{i}+\left(-x\right)\mathbf{j} \tag{02} $$ and $C$ is the semicircle of ...
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1answer
35 views

Vector Distance

let there be a line L: $\frac{x-1}{2}= \frac{y+1}{3}= \frac{z}{1}$ and a plane: $2x-y-z=5$. With this given data find: a line L1, such that L1 is parallel to L, is in P, and the distance between L and ...
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2answers
22 views

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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Any efficient approach to find the minimal vectors?

$\bullet$ A vector $X=(x_1,\cdots, x_m)$ is less then vector $Y=(y_1,\cdots,y_m)$ when $x_i\leq y_i$, for each $i=1,\cdots, m$, and for at least one $j$, we have $x_j<y_j$. $\bullet$ A vector ...
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0answers
11 views

Computing the normal vector when finding the flux

Use a parametrization to find the flux $\int \int_S F\cdot n$ $d\sigma$ across the surface in the given direction: $F=xy\overrightarrow i -z\overrightarrow k$ outward (normal away from the z-axis) ...
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0answers
25 views

How to solve this vector algebra problem.

This is a problem from a competitive exam. I tried solving it using properties of scalar triple product. I was only able to do this much From the first three terms I have taken $\vec{d}$ ...
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3answers
41 views

What is the difference between a linearly independent set and a set that spans $\Bbb{R}^m$?

This is more of a conceptual question. Here's what I know about a linearly independent set of vectors: A set of vectors $\{v_1, ..., v_p\}$ is linearly independent if the equation $$x_1v_1 + x_2v_2 + ...
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1answer
42 views

How to you find out what a matrix does to an equation.

Lets say I have an equation of a plane, $$x-3y+2z=0 $$ and I get matrix to transform it with say a 3x3 matrix with just a-i as place holders for the values in the matrix. How would I find what the ...
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0answers
23 views

Gram-Schmidt Process, finding orthonormal basis

Suppose I'm given $2$ random vectors $(v_1,v_2)$ I want to find orthonormal basis $(w_1,w_2)$ Are the following equivalent? for the $w_2$ case $$w_1=\frac{v_1}{\|v_1\|}$$ ...
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0answers
19 views

Cross Product of two perpendicular vectors

Say I have two perpendicular vectors $\bf a$ and $\bf b$, and any vector $\bf c$, can anything be said about $(\bf a \times \bf b) \dot \bf c$?
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2answers
29 views

What determines the direction of cross product resultant vector?

Why do we use the right hand rule to determine the direction of the vector resulting from using the cross product? A resultant vector that was directed in the opposite direction would also be ...
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1answer
24 views

How to determine if (1,0,1,1), (1,1,0,1) , (0,1,1,1) spans $R^4$?

I set up a system where $a(1,0,1,1) + b(1,1,0,1) + c(0,1,1,1) = (1,1,1,1)$ (the standard basis of R4) then i found that $a + b = 1$ $b + c = 1$ $a + b + c = 1$ which implies that $a = c = 0,$ and ...
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0answers
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How to find the appropriate weights to maximize the third coordinate while the first two are zeros

Let's assume, that $v_1, ..., v_n \in \mathbb{R}^3 $ and $ \lambda_1, ..., \lambda_n \in [0, 1] $ The $ v_1, ..., v_n $ vectors are given. I have to find the appropriate weights ($ \lambda_1, ..., ...
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+50

Geometric interpretation of adding dependence on a otherwise constant in a vector field.

So if i have $$ \vec{u} = \Omega r \vec{e_{\theta}}$$ Now if i take the curl $$\omega = 2\Omega\vec{e_z}$$ This is what we expect, we have a "rotating flow". Who's curl would be pointing in the z ...
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1answer
14 views

Surface Normal from Cross Product

Given an equation (in this case $x^2-y^2+z=0$) how would I find the surface normal using a cross product at a certain $(1,2,3)$ point? I know how do it with $grad(f)$ but I presume that isn't what ...
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26 views

How to formalise a procedure involving Cartesian products of sets of vectors and transformation in matrices?

I am asking for an help to formalise with the correct notation the following procedure. Let $n\in \mathbb{N}$. Let $\{0,1\}^{n-1}$ be the set of vectors of dimension $(n-1)\times 1$ with each ...
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2answers
36 views

Maximum value of $|(\vec {a} \times\vec {b} ).\vec{c}|$ is $\frac{A}{10}$

Let $\vec {a}$ and $\vec {b}$ be unit vectors. If $\vec{c}$ is a vector such that $\vec{c}+(\vec{c} \times \vec{a})=\vec {b}$ then maximum value of $|(\vec {a} \times\vec {b} ).\vec{c}|$ is ...
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2answers
17 views

Find unit vector that bisects two directed line segments. [on hold]

I'm trying to find the 2D unit vector that bisects two directed line segments with sign relative to the orientation of the line segments (left-hand side should be positive). Here is a graphic that ...
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1answer
15 views

Finding the inverse of a map given in vector form.

The question asks me to find the inverse map $ \mathbf\Phi^{-1} $, of: $$ \mathbf{\Phi}(\mathbf{x})= \mathbf{n\lor(x \lor n)} + \alpha\mathbf{(n \cdotp x)n} $$ for $\alpha$ such that the inverse ...
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3answers
25 views

Can p(1,2)= 1 and p(2,4)=3 be linear opperators?

This is a linear algebra question... So I know that the two conditions for linearity are additivity and homogeneity. Typically Ives seen examples where the ...
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0answers
17 views

How to reparametrize a geodesic such that $\nabla_V V = 0$

In Nakahara's section on geodesics he says that $\nabla_V V = fV$ can be reparametrized to give $\nabla_V V = 0$ if we use the reparametrize the tangent vector components $t \rightarrow t'$ such ...
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1answer
26 views

What is this vector equation?

I was going through some documents and I came across this vector equation (the vector is composed of a real part and imaginary part): if: $ v = a + j*b $ then: $w = \sqrt(|a|) * \sqrt(\frac{1 + ...
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1answer
25 views

How to setup vector story problems

I'm studying for my trig final and I know how to do all the math, but I don't always understand how to setup the story problems. Mostly I'm struggling with vector story problems. For example: Forces ...
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1answer
12 views

Calculate a point on a line at a specific distance to other point [closed]

How I can calculate a Vector3 in a line from A to B at a specific distance of C. Or it could be a line rotated until line intersects another if they can intersect. in a 3D space, I appreciate your ...
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0answers
23 views

Vectors and tractors [closed]

Chaz is using a rope tied to his tractor to remove an old tree stump from a field. Which method given below, a or b, will result in the greatest force applied to the stump? Assume that the tractor ...
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3answers
30 views

How to find an unknown matrix which when multiplied with a vector gives the cross product of 2 vectors

Question Image I am trying to find the matrix $[u]_x$ as shown in the image. $u \times v$ is easy to calculate but how do I find the matrix $[u]_x$ such that $[u]_x v = u \times v$ ?
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1answer
21 views

Find point on rectangle where vector intercepts [closed]

I have a vector in the centre of a rectangle pointing out of the rectangle. The size of the rectangle is known. The vector is known. The magnitude of the vector is always greater than the distance to ...
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2answers
44 views

Canonical notion of parallel transport

I have a "What is the right search term?" style question: Suppose $S\subset\mathbb{R}^3$ is a surface and that we are given two points $x,y\in S$. Furthermore, take $v_x\in T_x S$ to be a tangent ...
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27 views

Primordial elements of a vector space

We were given the following problem in our Algebra class. Let $V$ be a $K$-vector space (not necessarily finite dimensional), and fix a basis $(e_i)_{i \in I}$ of $V$. If $x = \sum \xi_ie_i \in V $, ...
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1answer
34 views

Using divergence theorem to calculate surface of sphere

I want to calculate: $$\iiint_V div (\overrightarrow F \cdot \space dV) $$ with $\overrightarrow F=x^3\hat i+ y^3 \hat j+z^3\hat k$ and Surface of sphere given as $x^2+y^2+z^2=r^2$ So, first I ...
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1answer
48 views

Vectors : Line of intersection no Z axis component?

I've been doing and exposing to many vectors questions, mostly from Singapore-Cambridge A level questions, and finding line of intersection between $2$ or $3$ planes, always yield line of ...
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1answer
24 views

Finding the Gradient of a Vector Function by its Components

In Multivariable Calculus, we can easily find the gradient of a scalar function (producing a scalar field) $f : \mathbb{R^n} \to \mathbb{R}$, and the gradient function would produce a vector field. ...
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20 views

Sum of element-wise product of multiple vectors

We know $\vec{a} \cdot \vec{b} = \sum{a_i b_i}$. Is there a way to express the sum $\sum{a_i b_i c_i d_i}$ in terms of four vectors $\vec{a}, \vec{b}, \vec{c}, \vec{d}$? UPDATE: Actually I'm trying ...
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1answer
14 views

Proving that point of intersection of diagonals of trapezium lies on the line passing through the midpoint of parallel sides?

Can someone tell me the steps to solve this problem? Prove by vector method that the point of intersection of diagonals of trapezium lies on the line passing through the midpoint of parallel ...
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1answer
23 views

Maximize triple product with respect to orthogonality contraint

I have the following problem: Suppose I have a plane $p$ defined by point $\vec{q}_1$ and normal vector $\vec{n}$. Also I have a line $g_2$, defined by point $\vec{q}_2$ and direction $\vec{l}_2$ ...
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3answers
44 views

Shortest Distance between planes

This is a question which puzzled our entire math class including our teacher, I'm referring to part (b), we're fine with part (a). We don't understand the reason for taking the dot product and the ...
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0answers
11 views

Inverse kinematics - How do i compute the du?

I am at the moment trying to implement at jacobian based inverse kinematics solver, which is given a current homogeneous Transformation matrix r(q) and a desired homogenous tranformation matrix ...
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1answer
29 views

Simplify vector equation $2\mathbf c - (\mathbf a + \mathbf b)\times(\mathbf a - \mathbf b)$

The unit vectors $\mathbf a$ and $\mathbf b$ are both perpendicular to a third unit vector $\mathbf c$. Additionally, a is at an angle of $\dfrac{\pi}4$ to b. Simplify the expression: $$2\mathbf c - ...
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i have posted a picture of the question . i can't do the second part and third part [closed]

enter image description herethe lines l1 and l2 have vector equations , I've shown that they intersect . i can't do the second and third part
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1answer
34 views

$O$ is a point inside cube such that $\vec{OA}+\vec{OB}+\vec{OC}+\vec{OD}=\vec{OM_1}$

Given a cube $ABCDA_1B_1C_1D_1$ with lower base $ABCD$ and upper base $A_1B_1C_1D_1$ and the lateral edges $AA_1,BB_1,CC_1,DD_1$ respectively. $M$ and $M_1$ are centres of the faces $ABCD$ and ...
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0answers
8 views

Compute a similarity matrix of documents from SVD (LSI)

I'm computing a SVD from a Matrix $m$ (columns = Documents ($D$) and rows = Terms ($T$) and truncate this matrix to lower the dimension of $m$ to $k$. From my resulting matrix $A = U \Sigma V^T$ I ...
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2answers
21 views

Is it possible to calculate the Xth percentile of a collection of wind vectors?

I have a collection of 30 weather model wind vectors (wind speed and direction) all valid at a specific point in space & time. It's fairly easy to break these apart and calculate a 75th percentile ...