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

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2
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1answer
25 views

If $x^{T}By = 1$, should $\operatorname{Tr}(Byx^{T}) = 1$?

would appreciate any hints with this question: Assume $x$, $y$ are both $n \times 1$ vectors, and that $B$ is $n\times n$. Given that $x^{T}By = 1$, should $\operatorname{Tr}(Byx^{T}) = 1$ ? Thank ...
0
votes
0answers
28 views

How to find the Cartesian equation of a plane in this example (in details)?

I'm solving an A Level paper, and came across this question. Basically, they have given plane $p$ has the equation $(\mathbf r-3\mathbf i)\cdot(2\mathbf i-3\mathbf j+6\mathbf k)=0$. Now, I can see ...
1
vote
1answer
21 views

How to find the vector equation of the line of the intersection of two planes $2x-y-3z=7$ and $x+2y+2z=0$?

We need to find the vector equation of the line of intersection of the above mentioned planes. I'm not that good with vectors so couldn't understand how to do it even though I had the answer in the ...
0
votes
0answers
17 views

Perpendicularity of plane and a cone. [on hold]

Show whether the plane $\:x+y+z=0$ cuts the cone $\:yz+zx+xy=0$ in perpendicular lines? Can anyone explain this in detail?
0
votes
1answer
18 views

Prove that the sum of the vectors from the centre to the vertices of a regular hexagon is 0

Prove that the sum of the vectors from the centre to the vertices of a regular hexagon is 0 Let's call the centre $O$ and the vertices are $A, B, C, D, E$ and $F$. Therefore, the sum in the ...
0
votes
0answers
13 views

Prove Bernoulli Function is Constant on Streamline

I have an incompressible, inviscid fluid, under the influence of gravity, with a velocity potential: $$ \mathbf{u} = (-\cos(x)\sin(y), \sin(x)\cos(y), 0) $$ Using Euler's equations, $$ \mathbf{u} ...
1
vote
0answers
36 views

Interpolation and mapping between scattered vectors in two unequally dimensioned spaces

Imagine two spaces: An ‘input’ space with dimension $m$. An ‘output’ space with dimension $n$. $m \geq n$ There are points in each of these spaces defined such that some characteristic is ...
0
votes
0answers
13 views

Finding closest vector for all rows in a matrix

I have two matrices 1. D ($m \times n$) and 2. C ($k \times n$). Typically, $m \approx 10^4, n \approx 100, k \approx 100 $. For each row r in D, I need to find the index of the row in C that's ...
0
votes
2answers
33 views

How to find a vector field that is perpendicular to the surface?

Im a bit confused with this question. Lets have the equation $z= x^2 + y^2$ therefore gradient f is perpendicular to surface $f=$ constant. In my case it would be $(2x,2y,-1)$ is perpendicular to ...
0
votes
1answer
25 views

inner product of two random vectors

Two random vectors $\mathbf a$ and $\mathbf b$. Vector $\mathbf a$ has uncorrelated entries satisfying $\mathbb E [\mathbf a \mathbf a^{\rm H}]=\sigma^2{\mathbf I}$. Now I need to calculate ${\mathbb ...
0
votes
2answers
29 views

Finding perpendicular vectors using dot product

Use the dot/scalar product to solve the problem Line 1 has vector equation $(2\mathrm{i}-\mathrm{j}) + \lambda(3\mathrm{i} + 2\mathrm{j})$ Find the vector equation of the line perpendicular to Line 1 ...
0
votes
1answer
14 views

Find the basis when integration is in the condition

Let $V$ be the set of all polynomial $f(x)$ in $P_2$ s.t. $\int_{0}^3 f(x) dx =3f(1) $ If $V$ is a subspace of $P_2$ find a basis of $V$. Can somebody help me get started? The integral condition ...
0
votes
0answers
8 views

Weighted average of unit vectors

I am trying to determine the yaw (z) axis of a vehicle given a bunch of sensors, each of them subject to different kinds of error. This means I get a bunch of unit vectors for the z axis, expressed ...
3
votes
1answer
31 views

Expressing a point in $\mathbb{R}^n$ as a sum of unit vectors

I'm pretty sure that any point in $\mathbb{R}^n$ can be written as a sum of finitely many unit vectors (in $\mathbb{R}^n$, of course). However, I have no idea how to go about proving this. Any ideas? ...
0
votes
0answers
24 views

Parametrisation of the curve after a short time

I am trying to wrap my head around this differential geometry problem. I am given velocity V with components in the principle normal and binormal directions. Then I am given an approximation of the ...
0
votes
1answer
19 views

Find the equation of a plane

The plane passes through $(3,2,-1)$ and $(1,-1,2)$ and is parallel to the line $v=(1,-1,0) + t(3,2,-2)$ I know how to do this with a vector perpendicular to the plane and that passes through a point, ...
0
votes
1answer
23 views

Linear independence of matrix-conjugate vectors

Let's define the vectors $\mathbf{v}_1,\dots,\mathbf{v}_m \in \mathbb{R}^n$, with $m\leq n$, to be mutually conjugate with respect to matrix $\mathbf{A} \in \mathbb{R}^{n\times n}$ if ...
1
vote
1answer
9 views

Geometric meaning of negative triple product

Create a tetrahedron with corners $A=(1,2,4)$, $B = (1,0,2)$, $C = (2,1,3)$ and $D = (4,1,1)$. Determine the angle $\alpha$ between the edge $AB$ and the side $BCD$. The first step I make is ...
0
votes
2answers
36 views

Why are these not bases in $\mathbb{R}^4$?

I know that bases vectors must span and be linearly independent. The (i) is not bases because the last vector contains $\pi$. The (iii) is not bases because they are not linearly independent. The ...
0
votes
0answers
24 views

Vector relationships with dot and cross product [on hold]

If $A, B$ and $C$ are $3D$ vectors in space $x,y,z$. Please prove the following relationship between dot product and cross product: $(A \times B) \times C = -A(B•C) + B(A•C) $ It was hinted in ...
0
votes
1answer
53 views

subset $W$ of $\mathbb{R}^3$ question

So the question says: consider the subset $W$ of $\mathbb{R}^3$ consisting of all vectors $$ \begin{bmatrix} x \\ y \\ z \end{bmatrix} \qquad \text{such that } x+y+z \geq -1 $$ Select all statements ...
0
votes
1answer
31 views

Find unit vector given Roll, Pitch and Yaw

Is it possible to find the unit vector with: Roll € [-90 (banked to right), 90 (banked to left)], Pitch € [-90 (all the way down), 90 (all the way up)] Yaw € [0, 360 (N)] I calculated it without the ...
0
votes
1answer
23 views

How to find a pair of basis vectors for the plane given by the equation?

Find a pair of basis vectors for the plane given by the equation x+2y+3z=0 I was given the Hint: you can find points on the plane by choosing values for two variables and solving for the third I am ...
0
votes
1answer
14 views

Sum of Linearly Dependent Vectors

Is it possible that vectors $v_1, v_2, v_3$ are linearly dependent, but the vectors $w_1=v_1+v_2$, $w_2=v_1+v_3$, $w_3=v_2+v_3$ are linearly independent? I believe the answer is no, this is not ...
1
vote
1answer
18 views

Writing unit vector r in terms of sine and cosine?

In my physics II class our professor has us go through through three steps to find the unit vector $\hat r$: Write the vector $\vec r$ $$\vec r=5\hat i + 5\hat j$$ Find the distance $r$ using ...
1
vote
0answers
8 views

Vector equation of line containing point and perpendicular to plane [duplicate]

How would one find the vector equation of the line that contains the point (x0, y0, z0) and is perpendicular to the plane Ax + By + Cz = D?
0
votes
0answers
30 views

How does one determine whether a coordinate basis is orthogonal or not?

Apologies for what is perhaps a very basic question, but I have been studying differential geometry with a view to gain a deeper understanding of general relativity and I have hit a stumbling block. ...
0
votes
1answer
20 views

application of vectors: sailboat floats in a current

A sailboat floats in a current that flows due east at 1 meter per second. Due to a wind the boats actual speed relative to the shore is $({\sqrt 3})$ meters per second in a direction 30 degrees North ...
-1
votes
1answer
26 views

Prove that $v$ is a linear combination of $v_1,…,v_n$ if it is a linear combination of $w_1,…,w_m$

How should I do this? Should I use the definition of linear independence?
0
votes
1answer
14 views

using law of cosines to find the equivalent vector

let there $\vec{A}=10cm$, $\vec{B}=15cm$ and the angle between them $25$ degrees, find the the equivalent vector $\vec{C}$ So we have: $|\vec{A}|=10$, $|\vec{B}|=15$, $\theta=25$ Using the law ...
0
votes
2answers
26 views

applications of vectors: woman in canoe

A woman in canoe paddles due west at 4mi/hour relative to the water in current that flows northwest at 2 mi/hour. Find the speed and direction of the canoe relative to the shore. I drew a diagram but ...
2
votes
2answers
20 views

Applications of vectors: Net force

Net force. Three forces are applied to an object, as shown in the figure. Find the magnitude and direction of the sum of forces. So I broke each force into its horizontal and vertical components and ...
0
votes
1answer
23 views

application of vectors: parachute in the wind

Parachute in the wind. In still air, a parachute with a payload would fall vertically at terminal speed of 4 m/s. Find the direction and magnitude of its terminal velocity relative to the ground if it ...
0
votes
0answers
36 views

Find scalar from a vector.

Given a vector $s = (s_1,s_2,s_3,s_4) \in \mathbb C^4$, find scalars $c_1, c_2, c_3$, and $c_4$ such that $s = c_1u_1 + c_2u_2 + c_3u_3 + c_4u_4$. One can obtain the column vector $c$ by multiplying a ...
1
vote
2answers
15 views

Angle vector in polar system represented by Cartesian vector

$x=r\cos\theta,\,y=r\sin\theta\implies r^2=x^2+y^2,\,\theta=\arctan(y/x)$ I can show that $\hat{r}=\cos\theta\hat i+\sin\theta\hat j$, where the hat vectors are ...
-5
votes
0answers
45 views

Prove the following statements about linear dependency in vectors spaces [closed]

Let $B = \{\vec{v_1}, \dots, \vec{v_k}\} \subset \Bbb R ^n$. Prove that if $k < n$ then there exists $\vec{v} \in \Bbb R^n$ such that $\vec{v} \notin \text{Span } B$. Prove that if $k > n$ ...
0
votes
0answers
12 views

Most Efficient clustering for vectors

I am trying to learn unsupervised learning in Python. My data is stored in list of lists as follows: ...
-2
votes
1answer
51 views

Prove or disprove these statements. [closed]

I have this statement and I need to prove or disprove it. Any help is appreciated. (1) Is it possible for solution set of a system [A| $\vec{b}^.$] of three equations and three variables, and ...
0
votes
1answer
25 views

What is the difference between base vectors and vector components.

I understand that you add them to create a vector. But are these just alternate names? In a math investigation I am using the two of them and I want to stay consistent in the way i name things. Thank ...
0
votes
0answers
28 views

Vector formula for the distance from a point to a line

I am seeking a proof that the distance from a point $\,\mathbf a\,$ to the line joining points $\,\mathbf b\,$ and $\,\mathbf c\,$ is given by $$\frac {|\mathbf a \times \mathbf b + \mathbf b \times ...
3
votes
4answers
59 views

Textbook for Vector Calculus

Can anyone recommend a textbook for studying vector calculus (vector analysis) only, that focuses on the theoretical mathematics behind vector calculus? Currently, I am using vector analysis by ...
0
votes
0answers
21 views

Calculating the Constant of Integration in Parametric, Vector-based Equations

I'm having trouble finding the constant of integration in parametric, vector-based equations. Given an equation: $$ a(t)\ =\langle \cos(t),\ \sin(t)\rangle $$ and $$ \int\ a(t)\ dt\ =\langle 0,\ ...
1
vote
2answers
49 views

magnitude of vector in algebra

I am trying to solve the following equation for x, in plain algebra this was easy $ y = x - \frac{1}{ x} $ $ x^{2} - yx - 1 = 0 $ $ x = \frac{-y \pm \sqrt (y^{2} + 4)}{2} $ However, throwing ...
3
votes
2answers
55 views

Pizza Delivery along the shortest path

You are the Captain of the USS Gauss and you have been flying in the direction $2 i + j + k$ for quite awhile. You are currently at the point $(6, 3, 3)$. Your helper monkey Mojo just ...
1
vote
1answer
35 views

A multiplication of vectors

From page 219, 'Machine Learning' by Murphy: ...
0
votes
0answers
13 views

Relations between unit vectors in polar and curvilinear coordinates

I'm a bit confused about the relations between unit vectors in curvilinear and polar coordinates in a plane. Let $u_{R}$ be the radial unit vector in polar coordinates and $u_{N}$ the normal unit ...
0
votes
0answers
25 views

Set, n-Tuple, Vector and Matrix — links and differences

I know this question has been asked like 1000 times, however all supplied answers were not really satisfying to me. My question concerns the similarities and differences between these mathematical ...
3
votes
2answers
47 views

Why “r” for vector-valued functions?

Why is the letter r typically used to describe vector-valued functions, like $r(t)$ or sometimes $R(t)$? Is it short for something like vector?
0
votes
0answers
18 views

Notation for Line Segment vs. Directed Line Segment

This may be nit-picky, but I noticed inconsistencies in a high school math text I was reading, and I'm curious what the world thinks. For the most part throughout this textbook, notation is used as ...
0
votes
1answer
25 views

Finding a 3rd point in a 3D triangle with known plane, two points and lengths of each side

I have a very similar problem to the below question. right triangle in 3D space, vectors, line intersection? Rather than having the unit vector $A$ I have the lengths $i_2$ to $i_3$ and $i_1$ to ...