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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-3
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6 views

checking free positions to fill real entries [closed]

general vector spaces how do you check "free positions to fill real entries" in the subspace"
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1answer
1k views

Calculate velocity vector given position on sphere, heading, and pitch

I would like to simulate a satellite's orbit iteratively using Cartesian coordinates. I am using the center of the earth for $(0, 0, 0).$ Given a heading, pitch, magnitude, and initial position, I ...
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1answer
50 views

How is $|\Delta \bf A|$ equal to $2A \sin \frac{\Delta\theta}{2}$?

How did the equation $$|\Delta\mathbf{A}|=2A \sin \frac{\Delta\theta}{2}$$ come? Tried looking at other books too but couldn't understand. I'm not allowed to upload images directly so I'm sharing the ...
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0answers
22 views

The dimension of vector and matrix

While I has tried to understand the principle of the dimension of diverse images (4-dimension), I has been baffled the definition of vector dimension. Everyone explaining the dimension of images said ...
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0answers
36 views

Dot product interpretation as an area

Edit: as per the comments below, the interpretation discussed in this question only makes sense if both vectors have strictly positive coordinates. I've been looking at the dot product between two ...
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2answers
17 views

Find the angle between the intersection of the two planes defined by $l_1=2x+y-z$ and $l_2=x+y+2z$ and the positive direction of $x$ axis .

Find the angle between the intersection of the two planes defined by $2x+y-z$ and $x+y+2z$ and the positive direction of $x$ axis . I know to find the angle between two planes we just need to find the ...
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How is $|vector ∆A|$ equal to $2A sin ∆(theta)/2$ [closed]

How did the equation $$|\Delta\mathbf{A}|=2A sin \frac{\Delta\theta}{2}$$ come? Tried looking at other books too but couldn't understand. I'm not allowed to upload images directly so I'm sharing the ...
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1answer
28 views

How to Find the $d^4p $ from the four vector?

Lets assume we are given the four vector of momentum P which can be written as: $$p = (p_0, p_t cos\theta , p_t sin \theta , P_L ) ........(1)$$ Where $P_L$ is the longitudinal competent. The ...
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7answers
244 views

Crux problem #33 with vector approach

On the sides $CA$ and $CB$ of an isosceles right-angled triangle $ABC$, points $D$ and $E$ are chosen such that $|CD|=|CE|$. The perpendiculars from $D$ and $C$ on $AE$ intersect the hypotenuse $AB$ ...
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1answer
86 views

Given the equation $\alpha \mathbf{v} + \mathbf{v}\times\mathbf{a} = \mathbf{b}$, solve for $\mathbf{v}$.

I'm reading a textbook at the moment that provides the following linear equation, $$ \alpha \mathbf{v} + \mathbf{v}\times\mathbf{a} = \mathbf{b}, $$ and asks to solve for $\mathbf{v}$. The form of $\...
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3answers
30 views

Find the coordinates of the vector x

Find the coordinates of the vector $x$, if it is known that is perpendicular to the vectors $a_1 = (4, -2, -3)$ and $a_2 = (0, 1, 3)$, forms an acute angle with a unit j and $|x| = 26$ (values must be ...
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0answers
26 views

Flux using Cartesian Coordinates

Find the flux of the vector field $\overrightarrow{F}=4x \hat{i} -2y^2 \hat{j}+ z^2 \hat{j}$ on the surface which is bounded by $z=0,z=3 $ and $x^2+y^2=4$. Here There are there surfaces that are ...
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2answers
24 views

Find the vector equation of the plane through any three points a, b,c

In $\mathbb R^{3}$, how to prove the plane through the points $\mathbf a$, $\mathbf b$, and $\mathbf c$ has the equation $$\mathbf r = (1-\mu-v)\mathbf a+\mu\mathbf b+v\mathbf c$$ I tried to evaluate ...
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0answers
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Find a parameterization of the caustic [closed]

Can you please help me with this question? If $\vec r(t) = [-\sin(t); \cos(t)]$ is the boundary of a coffee cup and light enters in the direction $[-1; 0]$, then light focuses inside the cup on a ...
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3answers
54 views

Proving that $\vec{r'}(t)$ is orthogonal to $\vec{r''}(t)$

With a given nonzero vector $\vec{r}(t)$, how do I that $\vec{r'}(t)$ is orthogonal to $\vec{r''}(t)$? The length ($||\vec{r'}(t)||$ is constant.) This is what I have tried so far. Let $\vec{r}(t)= &...
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3answers
1k views

Common perpendicular of two straight lines

I have the straight lines: $$d_1: \frac{x-1}2=\frac{y-3}1=\frac{z+2}1\\[4ex] d_2: \dfrac{x-1}1=\frac{y+2}{-4}=\frac{z-9}2$$ And I have to find the common perpendicular of these lines.
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2answers
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How to find the points of intersection of the perpendicular vector two skew lines

Correct me if im wrong, but this is what i know so far The cross product of two skew lines is the perpendicular vector between both those lines. The perpendicular vector intersecting two skew lines ...
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1answer
139 views

Using vectors how to prove $AB=AC$? [closed]

In a triangle $ABC$, $D$ is the mid-point of the side $AB$ and $E$ is the centroid of triangle $CDA$. If $\vec{OE}\cdot\vec{CD}=0$, where $O$ is the circumcentre of triangle $ABC$, then using vectors, ...
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0answers
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Can you find a 3-D vector to a point in space from a 2-D projection?

it's my first time asking a question on Math Stack Exchange, so if I haven't provided enough details, or if there's a better place to ask such questions, please let me know. I'm using a camera on a ...
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2answers
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how do i tell if a vector is parallel to another vector in R^6?

So far in my book I haven't learned any of the parallel or perpendicular notation.. so there must be some way to tell this answer that the book hasn't told me.. I looked back and there was nothing ...
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1answer
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Bounding the length of a sequence of same-dimension vectors over a finite integer interval such that the sequence contains no weakly decreasing pair

How long can a sequence of $i$-dimensional vectors become at worst if the components of each vector are natural numbers below $m$ and the sequence contains no pair of vectors such that the later ...
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1answer
60 views

Show for all real $u,v,w$: $\left|u+v\right|+\left|u+w\right|+\left|w+v\right|\le\left|u\right|+\left|v\right|+\left|w\right|+\left|u+v+w\right|$

Show that for all vectors $u,v,w \in \mathbb R^3$: $$\left|u+v\right|+\left|u+w\right|+\left|w+v\right|\le\left|u\right|+\left|v\right|+\left|w\right|+\left|u+v+w\right|$$ I know that we should show ...
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2answers
969 views

Prove by vector method that the diagonals of a rhombus bisect each other. Also, show that they bisect each other at right angles.

Prove by vector method that the diagonals of a rhombus bisect each other. Also, show that they bisect each other at right angles My Attempt: Let us consider a rhombus $OACB$ where $\vec {OA}=\vec {a}...
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2answers
42 views

Find coordinates of point in $\mathbb{R^4}$.

There is a problem here in Q. $5$ on the last page. It states to find coordinates of point $p$. Taking point $a=(3,2,5,1), \ b=(3,4,7,1), \ c= (5,8,9,3)$. Also, $b$ has two coordinates in common ...
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2answers
383 views

Point E and F are taken on the edges AD and BD respectively such that $E$ divides $\vec{DA}$ and $F$ divides $\vec{BD}$ in the ratio 2:1 each

The length of the edge of the regular tetrahedron D-ABC is $k$.Point E and F are taken on the edges AD and BD respectively such that $E$ divides $\vec{DA}$ and $F$ divides $\vec{BD}$ in the ratio 2:1 ...
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1answer
26 views

Compute orthogonal unit vector? [closed]

How can I find a unit vector orthogonal to $(\cos(45^{\circ}), \sin(45^{\circ}))$ in degrees? I read it has something to do with a cross product but I am unsure how to do it exactly; is there a ...
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0answers
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How to find normal vector given a point?

There is a ground surface which is uneven. Let's say you threw a ball and want to track its trajectory. You know the point where it hit the ground (x and y coordinates) and the slope of the ground at ...
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2answers
27 views

The angle between a line and a normal vector

The problem I am trying to solve is below: What is the angle formed by the line $(1,2,0) + t(-1,2,1)$ and a normal vector of the plane $x+y-z = 4?$ Give your answer in degrees. I am having a little ...
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1answer
30 views

Support Vector Machines (SVMs), unclear math steps

I am studing the maths behind the Support Vector Machines (SVMs), but there are two not clear steps. Following the video of 16. Learning: Support Vector Machines (MIT OpenCourseWare, minutes 14:24), ...
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2answers
113 views

Can we add a scalar and a vector? $a+b+\mathbf x = \mathbf y $?

I'm given the equation $$ a+b+\mathbf x = \mathbf y $$ With the vectors $\mathbf x=(x_1,x_2,x_3)$, $\mathbf y=(y_1,y_2,y_3)$ and the two scalars $a$, $b$. Is the following correct? As a vector ...
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0answers
19 views

Outer product of vector is zero

I want to show that if, for a column vector $v$ of dimension $(n \times 1)$; $$vv^T=\mathbb{0}$$ where $\mathbb{0}$ is the $(n \times n)$ zero matrix,then $v$ is necessarily the zero column vector, $\...
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0answers
15 views

Azimuth and Elevation to point from a ray with rotation [closed]

I've been wracking my head at this for a few hours now. I've been trying to find this for something I've been making for a game. Say there exists a target T within a 3D space with coordinates [xt,yt,...
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2answers
16 views

show that the lines are all in a plane and find the equation of the plane.

Three lines $L_1,L_2,L_3$ pass through the origin with the parallel vectors $$V_1=i+2j-k$$$$V_2=3i+5j+7k$$$$V_3=2i+3j+8k$$ are given,show that the lines are all in a plane and find the equation of the ...
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0answers
25 views

Principal Vector Exercise from Barrett O'Neill textbook

Prove that $\left|\begin{array}{ccc}{{v}_{2}}^{2}& -{v}_{1}{v}_{2}& {{v}_{1}}^{2}\\ E& F& G\\ L& M& N\end{array}\right|=0$ then $\overset{\to }{v}$ is principal vector where ...
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1answer
1k views

What is the difference between the orientation and the direction of a vector?

I came across this recently, and was confused regarding the difference between the orientation and the direction of a vector. Does the orientation refer to the relative coordinate which the vector is ...
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18answers
50k views

What is the difference between a point and a vector?

I understand that a vector has direction and magnitude whereas a point doesn't. However, in the course notes that I am using, it is stated that a point is the same as a vector. Also, can you do ...
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0answers
26 views

How to rearrange vectors in cross product? [duplicate]

I have an equation: $\vec{v}=\vec{w}\times \vec{r}$ How to I separate the $\vec{w}$ and write it in terms of $\vec{v}$ and $\vec{r}$? I tried to re-arrange this equation so that I could find the ...
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2answers
40 views

In the triangle $ABC$ with two given points $P,Q$ on the plane of the triangle, show that the points $P, Q $ and $C'$ are colinear.

I am given a triangle $ABC$ with the points $P, Q$ on the plane of the triangle such that: $$\overrightarrow{PC} = \dfrac{3}{2} \overrightarrow{BC} \hspace{2cm} \overrightarrow{AQ} = \dfrac{1}{4} \...
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0answers
29 views

Find missing point of Isosceles triangle by knowing all sides and angles in 3D

I have some problems finding the missing point of the isosceles triangle. I have known 2 points P(2,6,1) and Q(-1,3,5) and all sides and all angles of it. But when I'm trying to find another point O(x,...
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4answers
2k views

Vectors in XYZ Space with Negative Dot Product?

This question is from a Linear Algebra textbook by Gilbert Strang. I got the answer, but don't understand it complete. Can three vectors in the $xy$-plane have $\textbf{u} ⋅ \textbf{v} < 0$ and $\...
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0answers
14 views

Why do particles in Particle Swarm Optimization converge to the global minima when the local minima has just as much weight?

This is the formula for the velocity update in PSO: In simplified terms: velocity(n+1) = momentum_component + cognitive_component + social_component Now according ...
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1answer
22 views

Is it possible to rewrite these equation set to matrix form?

I am dealing with some equation set to calculate coefficients b in Matlab. The equation set is as below. The x y z are the inputs vector in size 4. b is output vector, size 4. ...
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2answers
51 views

Prove that a particle will never pass through the centre of a sphere under a condition.

Question: A particle was fired inside of a sphere. There was no gravity acting on the particle, no air resistance and each time it hit the inside of the sphere, it reflected without losing any ...
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0answers
27 views

Getting the coordinates of a point travelling a vector

So I'm currently having a problem with a python game. Let's say we have a square with a side of $600$ pixels, let's call the top segment $AB$ and the bottom one $CD$. Now on the $AB$ segment there is ...
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1answer
24 views

Prove that any collection of vectors that includes zero vector can’t be linearly independent.

I am stuck at the first step which is the definition of linear independence, that is, vectors $v_1, v_2, ......, v_k$ are linearly independent if and only if there exist scalars $c_1,c_2,.....c_k$ ...
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1answer
44 views

how to prove that spherical coordinates are orthogonal using cross product in cartesian?

The principle which I have been taught is taking $d\hat{\phi}$,$d\hat{\theta}$ and $d\hat{r}$, and the cross product of any two for example $d\hat{\phi}$ and $d\hat{\theta}$ should give me vector ...
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0answers
43 views

Curly braces notation with vectors as elements

In a paper by Camescasse et al.: "Bistable buckled beam: Elastica modeling and analysis of static actuation" in equation 23, they use a (to me) unknown notation for which i can't find any ...
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1answer
28 views

How to find the equation of the 3D plane passing through three points? [closed]

My solution as follow Normal(N) did not fit and take abbreviation like(A,B,C) to more simple showing. I wonder whether be true. $ $$\bf {P_1(x_1,y_1,z_1), P_2(x_2,y_2,z_2), P_3(x_3,y_3,z_3)}$ $$U=\...
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4answers
1k views

Vector equation for a cone

I’m not sure how to proceed with the following question. Could anyone help me? Thanks. Write down a (vector) equation for the right-angled cone centered around the line $$x = y = z$$ in three-...
0
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1answer
9 views

Find vector with specific angles

$ v = [{1, 1, 0}] $ and $ w = [-1, 0, 1] $. Find such vector $ u \in \langle v, v \times w \rangle $, that: $$ 1) \space \space \alpha(u, v) = \frac{\pi}{3} \\ 2) \space \space \alpha(u, v \times w) =...

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