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

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0
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0answers
25 views

Is “angle between two directions” appropriate?

I know vectors have both a magnitude and a direction, and I know that one may calculate the angle between two vectors. I am reviewing an academic paper where one of the author has written " This is ...
1
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2answers
32 views

Differentiate vector function wrt vector

I have a function $\frac{df(\mathbf{y})}{d\mathbf{y}}=\mathbf{y}g(\kappa)$ where $\kappa=||\mathbf{y}||_2$ and $g(\cdot)$ is a scalar function. Thing is when I differentiate this function I get a ...
5
votes
4answers
62 views

Given vector $\vec x = \left\{ x_i\right\}_{i=1}^n$ find an algebraic expression for $\vec y = \left\{ x^2_i\right\}_{i=1}^n$

Given vector $$\vec x = \begin{bmatrix} x_1 \\ \vdots \\ x_n \end{bmatrix},$$ How can we write out vector $$\vec y = \begin{bmatrix} y_1 \\ \vdots \\ y_n \end{bmatrix} := \begin{bmatrix} x^2_1 \\ ...
1
vote
2answers
17 views

Div$f$ is invariant under an orthogonal change of coordinates

Let $f: \mathbb{R^n} \to \mathbb{R^n}$ and $Df$ exists. I need to show that div$f$ is invariant under an orthogonal change of coordinates. Let $T:\mathbb{R^n} \to \mathbb{R^n}$ be an orthogonal ...
1
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2answers
24 views

Closed under scalar multiplication [on hold]

The subset of $\mathbb{R}^2$: $\{ (x,y)| y=\frac{7}{2}x\}$ is a subspace of $\mathbb{R}$. How can I prove that the subspace is nontrival ?
2
votes
2answers
25 views

Show that any 2D vectors can be expressed in the form…

(a) Show that any 2D vector can be expressed in the form $s \begin{pmatrix} 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 7 \end{pmatrix},$ where $s$ and $t$ are real numbers. (b) Let $u$ and $v$ be ...
0
votes
1answer
14 views

Convert two orthogonal unit vectors into euler vector

I have two orthogonal unit vectors that would correspond to an orientation of the Z and Y axes. I want to convert this to a rotation/Euler vector. In other words, I want to convert between two ...
-1
votes
2answers
20 views

relative velocity o level

A boat can travel at 3.5 m s in still water. a river is 80 m wide and the water flows at 2 m/s. calculate 1. the shortest time to cross the river and the distance down stream that the boat is carried ...
1
vote
4answers
29 views

Equation perpendicular to 2 non-parallel planes

Good day sirs! Can you help me with this questions? Find the general equation of the plane: (1) Through $(3,0,-1)$ and perpendicular to each of the planes $x-2y+z=0$ and $x+2y-3z-4=0$ (2) ...
0
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0answers
6 views

Find curl of vector field in 2Space implicitly in 3Space

I have a vector in 2Space, which (I'm assuming I need to implicitly treat as if it's in 3Space due to the fact curl is a cross-product which needs vectors to be in 3Space. $$ \text{ For each of the ...
2
votes
1answer
32 views

Is taking the real part required in vector orthogonality and projection?

In a real inner product space, two vectors are orthogonal if $\langle \mathbf{u}, \mathbf{v} \rangle = 0$. Similarly, $$\operatorname{proj}_{\mathbf{u}}(\mathbf{v}) = ...
0
votes
1answer
27 views

How to differentiate with respect to component of a vector?

Let $\vec{\alpha}=\frac{m(\vec{x})}{x^2}\vec{x}$ where $\vec{x}=(x_1,\,x_2)$. In a book I read in Eq.(3.24), it was given that $$ \frac{\partial \alpha_1}{\partial x_1}=\frac{d m}{d ...
2
votes
1answer
24 views

Significance of the derivative of a scalar field

I read somewhere that if the temperatures of all points of a huge room were plotted then the derivative at a certain point would give a vector whose direction points in the direction of the hottest ...
0
votes
1answer
22 views

How do I take the derivative of this vector valued function?

Problem: Find the velocity at time $t$ of the particle whose position is $\hat{r}(t)$: \begin{align*} \hat{r} = e^{-t} \cos(e^t) \hat{i} + e^{-t} \sin(e^t) \hat{j} - e^t \hat{k} \end{align*} This is ...
-1
votes
1answer
18 views

Direction vectors and cross product

I need some help with a quick question that I'm doing on an online math course regarding vectors. The question is: Based on the diagram in Question 1 or your previous answers, determine the cross ...
2
votes
2answers
31 views

Using the cross product to find the angle between two vectors in $\Bbb R^3$

Let $$u = \langle 1, −2, 3 \rangle \qquad \text{and} \qquad v = \langle −4, 5, 6 \rangle$$. Find the angle between $u$ and $v$, first by using the dot product and then using the cross product. ...
0
votes
1answer
21 views

Understanding step in deriving the formula of the curvature.

Last formula on third page of the document: Computation of $\vec{r'}(t)\times \vec{r''}(t)$ From the previous two formulas and using the properties of cross products we see that ...
-1
votes
1answer
12 views

How to find the bearing and velocity of a boat on a flowing river

Point A is on the west bank of a river and point C is directly across from it on the east bank. The river is 648.6 meters wide and flows south at 2.45 km/hr. A boat wants to cross the river from point ...
1
vote
2answers
50 views

Spacecurve of intersection between surfaces

Problem The spacecurve of intersection between the surfaces $opp$ & $\alpha$ (above z=3) has to be found, i.e. the intersection of the blue and green surface, above the red pane (z=3). Plots ...
3
votes
8answers
88 views

For two vectors $a$ and $b$, why does $\cos(θ)$ equal the dot product of $a$ and $b$ divided by the product of the vectors' magnitudes?

While watching a video about dot products (https://www.youtube.com/watch?v=WDdR5s0C4cY), the following formula is presented for finding the angle between two vectors: For vectors $a$, and $b$, ...
0
votes
1answer
29 views

Rate of change for no stretch/compression

I am reading about cloth simulation from here. This is what one of the part says - Shouldn't the condition for no compression/stretching be Wu = 0 If there is no stretch/compression along ...
0
votes
1answer
16 views

Derivative of vector and vector transpose product

I saw this answer here : Vector derivative w.r.t its transpose $\frac{d(Ax)}{d(x^T)}$. I am finding difficult to understand the part in red. What rule is that ? If I apply multiplication rule, ...
1
vote
1answer
9 views

Let $\vec F(x,y)=(y+xg(x),y^2), \vec F(1,1)=(3,1)$. $\vec F_x \perp \vec F_y$.Find $g$.

Let $\vec F(x,y)=(y+xg(x),y^2), \vec F(1,1)=(3,1)$. $\vec F_x \perp \vec F_y$ Find $g$. Attempt: I look for the partial derivatives, I did so differentiating each coordinate with respect to ...
-3
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0answers
39 views

Vector spaces and nontrivial subspace. [closed]

Give an example of a subset of $\mathbb{R}^2$ that is a nontrivial subspace of $\mathbb{R}^2$? $\mathbb{R}^2$ as $\{(a, b) \mid a, b \in \mathbb{R}\}$
0
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0answers
23 views

Taking derivate wrt a vector

I'm trying to read through Wiki's description of the Levenberg-Marquardt algorithm. I've taken linear algebra, but I've always been fuzzy about taking derivatives with respect to a vector and just ...
0
votes
4answers
40 views

How to find perpendicular vectors in 3D

Find all values of a such that the vector $q = \langle 2, a, –2\rangle$ is perpendicular to the vector $p = \langle –3, a, 5 \rangle$.
0
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0answers
19 views

How to multiply the elements within a vector using matrix operations (e.g., dot product)?

Suppose a vector $\vec{v}^T=(v_1, v_2, \ldots, v_n)^T$. To sum the elements within the vector, I can use the dot product with a column vector of ones, $\sum_i v_i = \vec{v}^T \cdot \vec{1}$. My ...
0
votes
1answer
16 views

$|\vec{r}-\vec{r}_1|=\frac{1}{2}|\vec{r}-\vec{r}_2|$ is the equation of a sphere?

I am told that the set of all $\vec{r}$ for which $|\vec{r}-\vec{r}_1|=\frac{1}{2}|\vec{r}-\vec{r}_2|$ is true forms a sphere---however, my semi-intuitive reading of this equation puts a "weaker ...
0
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0answers
20 views

Intersection of two moving objects in 3D

There are two objects, where the known data is the position and velocity in 3D vector format. I`m interested in the time and position of the intersection between the two, and possibly without ...
0
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0answers
18 views

Notation sumation confusion

I am reading paper about additive schwarz preconditioner, where following notation is used in order to obtain matrix C $$C_i = \sum_k (I^k B^k (P^k u_i)R^k)$$ . My question is, what's dimension of ...
-1
votes
1answer
32 views

Vectors expressed as combo of two other vectors [closed]

Let $u$ and $v$ be non-zero vectors. Show that any two-dimensional vector can be expressed in the form $su + t v$ where $s$ and $t$ are real numbers, if and only if of the vectors $u$ and $v$, one ...
-3
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0answers
17 views
1
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1answer
41 views

Vectors and representation of lines

Given vectors $p$ and $d$, we can describe the line through $p$ in direction $d$ as the vectors $x$ that satisfy $x = p + t d$ In this problem we explore another representation for lines. a) Let $p ...
-1
votes
0answers
15 views

3D Geometry-If alpha /2 ,beta /2 and gamma /2 are the angles with 3 axes.Then, cos alpha + cos beta + cos gamma [closed]

If alpha /2 ,beta /2 and gamma /2 are the angles which a line makes with the x,y,z axes respectively.Then, cos alpha + cos beta + cos gamma = ? a-> 1 b-> (-1) c-> 2 d-> 3
1
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0answers
52 views

How can I use the Bullet-Physics's ray-cast normal to calculate angles for a object to lay on a surface?

[Give the normal of a surface in XYZ format, how do I calculate rotations (also in XYZ format) needed to set an object parallel to the surface?] I have a collision library that uses the bullet ...
1
vote
3answers
48 views

Finding dimension of subspace

I know that any polynomial in subspace $W$ must have $(x-1)$ as factor so that $p(1)=0$ But I don't understand how $p'(2)=0$ can be incorporated. Thankful for any kind of help.
-1
votes
1answer
23 views

Flip Normal of Plane

Hi If I have a general form of a plane such as : $2x + 4y - 6z + 10 = 0$ what is the method for inverting the normal vector?? I realize the normal vector for this equation is $(2, 4, -6)$ would ...
0
votes
1answer
20 views

Show that the ratio of the distance $BD$ to $DC$ is $\gamma:\beta$

I'm struggling on the following question My approach to the question is as follows: The equation of line $BC$ is: $\lambda_1$b+$(1-\lambda_1)$c The equation of line $AP$ is: ...
1
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1answer
12 views

How to divide a vector on a sphere into northern and southern components?

Suppose we have $S^2$ and a vector $\vec{A}$ pointing at a random direction. Let us divide the sphere into $S_N$ for $0 \leq \theta \leq \frac{\pi}{2}$ and $S_S$ for $\frac{\pi}{2} \leq \theta \leq ...
1
vote
1answer
52 views

Proving the Non-existence of an Orthogonal Vector in $\mathbb{R}^n$

If $X$ is vector in $\mathbb{R}^n$ with all components > 0 then is it true that a non-zero vector, $Y$, with all components ≥ 0, can not be orthogonal to $X$ ? Considering the angles that $X$ makes ...
0
votes
1answer
45 views

What is the vector equation of a straight line?

I know that, the vector equation of a line passing through $(x_0, y_0, z_0)$ is, $ \vec r = \vec r _0 + t \vec v $ Where, $\vec r$ = the vector for the subject line. $\vec r_0$ = a position vector ...
2
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2answers
43 views

How to prove the existence of vectors?

I solved this problem for a few specific vectors but I don't know how to prove this in general? Grateful for any kind of help!
1
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1answer
38 views

How to solve this vector equation for optical flow

I am unable to solve for $\textbf{h}$ in the following equation $\sum\limits_{\textbf{x}=1}^n2\partial F(\textbf{x})/\partial\textbf{x}(F(\textbf{x}) + \textbf{h}^{T}\partial F(\textbf{x})/\partial ...
-1
votes
1answer
39 views

Vector Alignment [closed]

Can anybody help me solve this problem? Provide a 4x4 Transformation Matrix aligning arbitrary vector $\vec{V} = (𝑣_𝑥,𝑣_𝑦,𝑣_𝑧)$ with arbitrary vector $\vec{U} = (u_x, u_y, u_z)$.
1
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1answer
7 views

How to find the direction of vector $ v(2)$ in this problem?

I'm having difficulties finding the direction for vector $v(2)$ in this problem: Given the vectors $v(1) = <3,4>, v(2) = <1,1>$, and $v(3) = <2,7>$ What is the direction of vector ...
0
votes
2answers
30 views

How to scale an unit vector $u$ in such way that $a u\cdot u=1$ where $a$ is a scalar

I have a problem of scaling a unit vector $u$ in such way that its scalar product (by itself) multiplied by a costant has to be equal to one. $$a u\cdot u=1$$ How do I do it? Edit: the unit vector - ...
0
votes
1answer
23 views

Findings the dot product between two non-adjacent vectors

I need to find $r\circ(p-q)$ from the below diagram, and since $r$ is prependicular to $p$, I only need to calculate $-(r\circ q)$ when I know that the modulus of $q=3$ and of $p=4$. Now, I know that ...
0
votes
1answer
17 views

Compare two different vector pairs

I have two different sets of vectors with the same dimension, $\dim = N$ where $N$ is around $300$. Assume a select a pair from set 1, $v_1$ and $v_2$. Then I select a pair from set 2, $u_1$ and ...
1
vote
1answer
29 views

Why isn't the square root is cancelled in this formula?

$\sqrt{\sum\limits_{i=1}^M \vec{V^2_d}(d)}$ This is the formula of the Euclidean length of a vector in the vector space. The vector $V$ has a power of 2 so it is $V^2$. Why isn't the square root of ...
-1
votes
1answer
49 views

Column vector of simultaneous equaations' solution

Struggling with some basics of Linear Algebra. Please help. Let's restrict the discussion to 2D space & consider the following simultaneous equations: $2x + 3y = 8, x + 2y = 5$ I understand ...