Use this tag for questions involving vectors, quantities that have magnitude and direction.

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0answers
19 views

Vectorial product to find cosine

We have A(2,1), O(0,0), B(0,3) and C(2,4). OACB is a parallelogram. Find the cosine of $\angle AOB$ I know that the cosine of an angle determined by two vectors can be written as ...
-3
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0answers
32 views

Proving volume of a general cone using Gauss theorem [on hold]

Please assist me with answering this question: The parametrization of a general cone is $$(x(t,\mu),y(t,\mu),z(t,\mu))=\left((1-\mu)C_1(t),(1-\mu)C_2(t),\mu h\right)\quad\quad 0\leq t\leq b,\quad ...
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4answers
40 views

How to use Cross Product Properites to do proof

How do I proceed with a proof for this question? Prove that: \begin{equation} (a \times b) \cdot (c \times d) = \begin{vmatrix} a \cdot c & b \cdot c \\ a \cdot d & b \cdot ...
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0answers
14 views

Define a curve from A to B and divide it into set of vector points.

I'm trying to define a curve from point A to point B and then divide it into set of vector points. I'm able to define a straight line from A to B and divide it, but I'm having problems with doing the ...
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3answers
23 views

Problem about planes

Say we have $2x+3y+3z=0$ which is a plane. Does that plane have infinite dimensions (it is a 2D "object" — forgive me as I am not a mathematician — but each side has infinite length) or is it just the ...
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2answers
36 views

Distance between point and plane - why use the dot product?

So according to this, the signed distance between a point and a plane will be the dot product of the plane's normal vector (does it have to be a unit vector?) and the point-in-plane minus the point ...
1
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2answers
38 views

Using Gram-Schmidt to compute the cross product of $3$ vectors in $\Bbb R^4$ [duplicate]

I want to ask about vector multiplication (cross product) in $4$-d. I heard that Gram-Schmidt process is involved but I am not sure how the process is involved. The multiplication involves $3$ ...
6
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1answer
51 views

Are there closed curves for which acceleration is orthogonal to position?

Can we find $\vec{f} : \mathbb{R}\rightarrow \mathbb{R}^3 $ such that $\vec{f}(t) \cdot \frac{d^2 \vec{f}(t)}{dt^2} =0$ and $\vec{f}(0) = \vec{f}(T)$ for some $T >0$ ? Exclude the trivial cases. I ...
0
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1answer
16 views

“Similarity” of two vectors

Imagine I have three vectors v1 = [1,1] v2 = [.9,.9] v3 = [.1,.1] I want to see how closely related two vectors are in both Magnitude and Direction So consider a hypothetical "similarity" function ...
0
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1answer
26 views

Symbol to denote length of geometric vector

I have seen both $\left|\vec{u}\right|$ and $\left\|\vec{u}\right\|$ when referring to the Euclidean length of a geometric vector $\vec{u}$. Which notation is preferred. Is it true that the latter ...
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4answers
55 views

Intersection of two planes, how to represent a line?

If we have two planes: $$4x-y+3z-1=0$$ $$x-5y-z-2=0$$ and if we want to find a plane which contains the origin point and the intersection of the two planes given, how do we do it? What my teacher did ...
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3answers
29 views

cross product of vector and direction

We know that cross product gives a vector that is orthogonal to other two vectors. Let this vector denoted by $$|\vec{v} \times \vec{u}| = \vec{n}$$ Then $$\vec{n}\cdot \vec{u} = 0 $$ Everything okay ...
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2answers
28 views

Find 3D distance between two parallel lines in simple way

Is there a simple way to get 3D distance between two parallel lines given end points of each line?
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0answers
32 views

Is there a relation between vectors on these two spaces?

I've been reading lately one paper on Physics, which basically presents one gauge theory approach to the problem of swimming at low Reynolds number. I've been trying lately to rewrite some of the ...
0
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1answer
8 views

Point within a Cube in 3D environment

I have a cube in 3D space with 8 corner points with their X,Y,Z Coordinates. I know how to test if any given point lies inside a cube by Comparing their coordinates to be greater or smaller than the ...
1
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2answers
29 views

How to scale a 2D vector and keep direction

I want to take any vector in R2 and scale its length to 1 while keeping the original direction (ratio of x component to y). As an example of my goal, let's say I have the vector (1,1), it would become ...
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2answers
36 views

A question on convex hull

Let $a_1, a_2,\ldots, a_n$ be $n$ points in the $d$-dimensional Euclidean space. Suppose that $x$ is a point which does not belong to the convex hull of $a_1, a_2,\ldots, a_n$. My question is, does ...
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1answer
30 views

Why is this triple product zero?

The triple product is given as:- $$ \vec A \cdot(\vec B \times \vec A) $$ $$ \vec A \cdot((-AB\sin\theta))$$ $$ -A\cdot AB\sin\theta \cdot \cos\theta$$ But the book says the answer is zero, I tried ...
1
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1answer
20 views

Finding the projection of a 3D vector along the direction of (i-j)

The question is:- Find the component of vector $$ \vec A = a_x\hat i + a_y\hat j + a_z\hat k$$ along the direction of $$ (\hat i - \hat j)$$ Since both the vector should be in the same dimensions ...
0
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0answers
18 views

Vector in vector notation

I'm a bit confused as far as notationally differentiating between row and column vectors goes. Suppose I define a column vector $$\boldsymbol{a} = (a_{1}, a_{2})^{T}$$ and another column vector ...
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1answer
29 views

Relation between dot product and magnitude of vector product [on hold]

Let $A$ and $B$ be two vectors in $\mathbf R^3$. Calculate the angle between the two given vectors if $A \cdot B= \left| A\times B\right|$. Please guide me how to find the answer of the above ...
0
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1answer
14 views

For which value of m are these 3 vectors linearly dependent?

In one of my revision worksheets there is a question which goes as follows: The vectors u=mi+j+k, v=i+mj+k and w=i+j+mk, where m is a real constant, are linearly dependent for either m=0, m=1, m=2, ...
-3
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0answers
22 views

Vector Geometry Proof with a Pentagon [on hold]

Pentagon $ABCDE$ is inscribed in a circle. For any edge of $ABCDE$, we can draw the line perpendicular through that edge that contains the centroid of the remaining three vertices. Show that these 5 ...
0
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1answer
30 views

Find angles between sides of triangle and coordinate planes ($xy,yz,zx$ planes) using three 3d vectors .

Given the following, three vectors: \begin{align*} \vec{a}& = 3i−2j+5k, \\ \vec{b}& =i−6j+6k, \\ \vec{c}& =2i+3j−k, \\ \end{align*} find the angles between sides of triangle and ...
-1
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1answer
28 views

Select certain elements from a vector array MATLAB [closed]

I am new to MATLAB. I have an array consisting of real numbers. I want to select the elements that are closest to integer values. I want one element per integer. I am using the "if" statement and ...
0
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0answers
33 views

Best way to quantify the difference between two vectors

There are plenty of ways of showing an error, or rather a deviation, between two vector quantities. What is the best choice? Specifically, at every timestep, I am comparing two vectors of curvature ...
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0answers
27 views

Function that maps all vectors to the origin?

I need a function that will map any vector (and any point on that vector) in the cartesian plane to (0,0) using only addition and subtraction. Is this possible?
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2answers
20 views

Relationship between vectors in convex quadrilateral

We have the triangle $ABC$. $M$ is the center of the line segment $BC$. $D$ is a point in the triangle's plane so that $ABDC$ is a convex quadrilateral. $N$ is center of the line segment $AD$. ...
0
votes
1answer
19 views

Stoke's Theorem Application on Cylinder

This is a question regarding Stoke's theorem's application. This is in regards to a problem from MIT OCW. My question is, referring to the answer provided, what closed surfaces are used in the proof ...
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1answer
19 views

Points on two skew lines closest to one another

Given two skew lines defined by 2 points lying on them as $(\vec{x}_1,\vec{x}_2)$ and $(\vec{x}_3,\vec{x}_4)$. What are the vectors for the two points on the corrwsponding lines, distance between ...
2
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1answer
39 views

Finding a basis for $V, W, V+W$ and $V \cap W$

Problem: Let \begin{align*} V = \left\{(x,y,z,u) \in \mathbb{R}^4 \mid y+z+u = 0 \right\} \end{align*} and \begin{align*} W = \left\{(x,y,z,u) \in \mathbb{R}^4 \mid x+y = 0, z = 2u \right\} ...
0
votes
2answers
54 views

Find perimeter and angle of triangle using three 3d vectors .

Given the following, three vectors: $$\vec{a} = 3\mathrm{i} - 2\mathrm{j} + 5\mathrm{k}\\\vec{b} = \mathrm{i} - 6\mathrm{j} + 6\mathrm{k}\\\vec{c} = 2\mathrm{i} + 3\mathrm{j} - \mathrm{k},\\$$ find ...
3
votes
2answers
57 views

Rotate a unit sphere such as to align it two orthogonal unit vectors

I have two orthogonal vectors $a$, $b$, which lie on a unit sphere (i.e. unit vectors). I want to apply one or more rotations to the sphere such that $a$ is transformed to $c$, and $b$ is transformed ...
2
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1answer
24 views

Find the equation of a plane when it passes through two points and parallel to two vectors

Q) A plane passes through points (1,1,-2) and (3,2,0) and is parallel to vectors i+2j+k and 2i-j-2k. Find the vector equation of plane in scalar product form. So, what I've done so far is take the ...
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2answers
38 views

Number of vectors over a finite field that are linearily independent to a subspace

let $S$ be a vector space over a finite field of size $q$ and let $T$ be a subspace of $S$. I am looking for a formula or an algorithm to compute the number of vectors from $S$ that are independent ...
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0answers
29 views

The equivalent of least squares, but for vectors

Given a set of poins, one can use a fitting method such as least squares to find the straight (or the parabola, or the 3rd grade equivalent) that's closest to all points at the same time (via ...
0
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1answer
30 views

How do you call a vector of length $n$, with all values equal to $\frac{1}{n}$?

Is there a specific name for a vector of dimension $n$, with all values equal to $\frac{1}{n}$? So, a vector that looks like this: $\vec{v} = \underbrace{(\frac{1}{n}, \frac{1}{n}, ..., \frac{1}{n}, ...
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2answers
59 views

Is u • v equal to |u • v|?

The following is a homework problem from an introductory course in linear algebra with the instructor's answer: Question: Use the geometric description of the dot product to verify the Cauchy ...
2
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1answer
28 views

Direction of rotation to transform from Point a to b on a unit sphere

If I have two points $a$, $b$, on a unit sphere, I believe I can determine the angle between them, expressed as vectors, as follows: $$\theta = \arccos\left(\frac{a\cdot b}{\|a\| \|b\|}\right)$$ ...
0
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2answers
29 views

What does cross product mean in simple words?

Two numbers $3$ and $4$ their multiplication is each one from the first number is repeated a number of times as the second number i.e. $3$ times $4$ is $(1+1+1)$ times four meaning $1+1+1+1 + 1+1+1+1 ...
2
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2answers
45 views

Divergence of inverse square vector field

After trying to get the divergence of a vector field I got: $${\nabla \cdot \Bigg(\frac{\vec r}{|r|^3}\Bigg)=\frac{\partial}{\partial r}.\frac{1}{r^2}=-\frac{2}{r^3}}$$ But in wikipedia found that ...
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1answer
15 views

Help describing a graph for vector valued functions

Doing a little summer study and my textbook doesn't have much answers so thought I'd ask here. The topic is an introduction to vector-valued functions The question asks to 'describe the graph' $\vec ...
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2answers
30 views

Proof with 3D vectors

Let ${a} = \begin{pmatrix}x_a\\y_a\\z_a\end{pmatrix}$, ${b} = \begin{pmatrix}x_b\\y_b\\z_b\end{pmatrix}$, and ${c} = \begin{pmatrix}x_c\\y_c\\z_c\end{pmatrix}$. Show that $(x_a,y_a,z_a)$, ...
0
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1answer
36 views

What is $\ker (4,-4)$

I found this equation in a textbook: $$\ker (4,-4)=\langle(1,1)\rangle$$ It's the first time I see this operator and couldn't find a description of what's happening here. Could someone help me out?
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1answer
45 views

How can i strictly prove this conclusion

$a,b\in R^n$. For all the $x \in R^n$ that satisfy $x\cdot a \geq 0$, will also satisfy $x\cdot b\geq 0$. Show that there exist a non-negative real number $\lambda$, that makes $b=\lambda a$ sorry ...
3
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2answers
81 views

Let $ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$. Determine $S+S+…+S $.

Let $$ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$$ By the usual notation for sum of sets let $$ 2S\overset{\text{not}}{=}S+S=\{(x_1+x_2,y_1+y_2) \ | \ (x_1,y_1), ...
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1answer
40 views

Losing a dimension when finding intersection between subspaces

Let $F=\mathbb Z_3, V=F^4$. Let $U=sp\{(1,0,0,0),(1,0,1,0),(0,1,1,1) \} \\W=sp\{(0,0,1,0),(-1,1,0,1),(1,1,1,1) \}$ Find $dim (U\cap W)$ we have $v\in U \text{ and } v\in W$ so $v=v$ ...
1
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1answer
22 views

Symbol representing a vector composing of two vectors

I have a vector including two vector's elements. How do I simply represent a vector with elemental vectors. Formally, I have three vectors $x, a=(a_i), b =(b_i)$ and $x=(a_1, a_2, ...
0
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2answers
34 views

Components of a vector given three points?

I'm trying to begin these two questions (25a and 26a specifically) but am at a loss on where to begin: No information on points on the graphs is given. I know the components of a vector with start- ...
1
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1answer
32 views

Find corners of a square in a plane in 3d space

I am given two angles (similar to theta and phi in spherical coordinates) from which I can calculate a normal vector to a plane in 3d space. I am also given the center point of the square and the area ...