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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.

1
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1answer
9 views

Linear independence proof of sublist from a list of dependent vectors

Let $\lambda$ be an eigenvalue of A, if no eigenvector of A associated with $\lambda$ has a zero entry.Then proof every list of $n-1$ columns of $A-\lambda I$ is linearly independent. This problem is ...
0
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1answer
26 views

shortest distance between two vectors

Whilst working on vectors I have come across a lot of problems like this. I am able to work it out for the shortest distance from a vector to a point, but not from a vector to a vector. Here is my ...
4
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2answers
300 views

How do we really get the angle of a vector from the components?

Usually when people discuss getting the polar form of a vector $v$, they present the following two formulas: $$\text{Magnitude}(v) = \sqrt{x^2 + y^2}$$ $$\text{Angle}(v) = \arctan \left(\frac{y}{x} \...
0
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0answers
10 views

Uniform precession motion, relative derivation, poisson vectors.

Suppose that you have a gyroscope with revolution symmetry around a perpendicular axis $\bf{e}$ such that the inertia tensor of this gyroscope can be written: $${\bf{Jc}} = {A\bf{I}} + (\Gamma - A)\...
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0answers
17 views

How can I compute a distance function for two sets of vectors?

Let's say I have two sets of vectors, $A$ and $B$. The cardinalities of $A$ and $B$ are not the same, i.e. $|A| \ne |B|$ however, each of the vectors from either set are of the same size, i.e. have ...
0
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0answers
12 views

Distance a from point in R3 to a surface defined by a parametric curve and a radius function?

I'm interested in studying the class surfaces defined by: Take an arbitrary parametric curve f : {0..1} -> ℝ3. Pick an arbitrary radius function ...
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0answers
16 views

Let $A$ be an improper orthogonal matrix… [on hold]

Let $A$ be an improper orthogonal matrix then adjoint of $A$ is equal to $$A$$ $$A^T$$ $$-A$$ $$-A^T$$
0
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2answers
39 views

Three equilateral triangles form a hexagon [on hold]

As I posted yesterday, I was learning about vectors yesterday. I know how to add and subtract them, but I can’t multiply yet. So here is an extra problem from my teacher I need help with: Given a ...
0
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2answers
73 views

Invertible matrix 2019

Let $A\in M_{n\times n}(\mathbb{C})$ be an invertible matrix. Show that there exist $u\in\mathbb{R}^n$ and $\lambda\in\mathbb{C}$ not null, such that $u=\frac{1}{\lambda^n}A^nu$ And I am not able to ...
2
votes
1answer
25 views

Two regular pentagons and the sum of vectors connecting their vertices

Today I was learning about vectors. The teacher gave me the following problem: Consider two regular pentagons $A_1A_2A_3A_4A_5, B_1B_2B_3B_4B_5$ on the plane. The center of the first pentagon is $O_A$...
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2answers
31 views

Geometric Relationship between Two Vectors [duplicate]

Consider two column vectors such that $a = (1,2,3)^T$ and $b = (-3,3,-1)^T$. What is the geometric relationship between $a$ and $b$?
0
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1answer
11 views

Given vectors OA and OB calculate vector AB

So the teacher said it's like this OB - OA = AB and I don't understand it. Why aren't we doing this instead: OA - OB? aren't they the same thing?
0
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1answer
23 views

Angle between sum of vectors

Let $u,v$ and $w$ be vectors in $\mathbb{R}^n$ and let $\theta(u,w), \theta(v,w)$ and $\theta(u+v,w)$ represent the angle between each listed pair of vectors. Does it hold that one of the following ...
1
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1answer
10 views

Two statements regarding orthogonal unit vectors and orthogonal complements respectively

I am new to linear algebra, and I was confused regarding the following question. I would really appreciate it, if anybody could give some feedback... True or False? $\left(\frac{1}{\sqrt14},\frac{-2}...
2
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1answer
21 views

Can you take the dot product of a column vector and a row vector (i.e. a vector and a dual vector)

I recently learned about the definition of work, namely the one involving path integration. W=integral of (F.dr). In this case, F is a vector field and dr is a small segment of the path r being ...
-1
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2answers
61 views

Determine point of intersection or find the value of $z$

Let $L_1$ be the line passing through the points $Q_1=(4, −2, −4)$ and $Q_2=(5, −1, −5)$ and let $L_2$ be the line passing through the point $P_1=(−13, −12, 6)$ with direction vector $\underline{d}=[6,...
0
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3answers
28 views

Is the requirement that $u+v$ be in $V$ if $u$ and $v$ are in $V$ a valid axiom for the definition of a vector space? Seems to get skipped

For example, at: https://en.wikipedia.org/wiki/Vector_space There are 8 axioms that a qualify a set to be a vector space. My professor also gave us 8. However, a textbook I'm reading states 10 ...
1
vote
1answer
20 views

Why does a unit vector point in the same direction? [duplicate]

I know how to compute the unit vector $$\hat{\textbf{u}} = \left( \frac{u_1}{||u||} , \dots , \frac{u_n}{||u||} \right)$$ and I also know how to show that this will have length 1 by using the ...
0
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2answers
28 views

How to find orthogonal vector to an arbitrary 3 dimentional vector [duplicate]

Given a vector $\begin{bmatrix}a\\b\\c\end{bmatrix}$ what is a simple solution to find any vector perpendicular to it?
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votes
1answer
28 views

Null space of a rotation matrix

If we have a rotation matrix of the kind: link to the rotation matrix how do i compute the null space of this matrix? I know that to obtain the null space we need to write the matrix in echelon ...
0
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0answers
12 views

how to calculate wall thickness of a mesh?

based on similar questions on mesh volume, volume of a mesh can be calculated by following equation: volume = ((vec1 x vec2) . vec3) /6 where vec1, vec2, and vec3 are the vectors from origin to a ...
0
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2answers
17 views

What is the component of a cross product b along a [on hold]

What is the component of a cross product b** along a
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1answer
29 views

Coordinate-free proof that two points are diametrically opposed

Let $c$ be the center of a circle with radius $r > 0$ and let $a$ and $b$ be two points at the circle. If there exists $t \in [0,1]$ such that $c = (1-t)a + tb$, then $d(a,b) = 2r$. I'm working ...
1
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1answer
48 views

Condition for which $|z-z_1|^2+|z-z_2|^2=K\in\mathbb{R}$ represent a circle

Let $K\in\mathbb{R}$ and $z_1,z_2\in\mathbb{C}$. Prove that the equation $|z-z_1|^2+|z-z_2|^2=K$ represent a circle iff $K\geq\frac{1}{2}|z_1-z_2|^2$. My Attempt $$ |z|^2+|z_1|^2-2\mathcal{Re}(...
0
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2answers
9 views

How do you evaluate vector magnitudes which include multiplication?

For the vectors u= (1,3,0) and v=(3,0,2), how do I find the magnitude of u+4v? For letters u and v, I try adding all 3 components of the vectors, squaring each component, then square rooting it. For v,...
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0answers
40 views

Find the vector equation for a line that passes through $P$ and intersects $L$

I just need to know if my answer is right. If it isn't please tell me what the answer is and what I did wrong. Question: Let $L$ be the line with parametric equations \begin{align*} x & = 5−...
1
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2answers
38 views

Can $\| \left( X'X + \lambda I \right) ^{-1} X'y \| = t$ be solved for $\lambda$?

In this post I suggested that the expression $$ \| \left( X'X + \lambda I \right) ^{-1} X'y \| = t $$ couldn't be easily solved for $\lambda$, because you need to "invert" the norm. But in general ...
0
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2answers
53 views

Find k when L2 passes through the P1 with direction vector

Q: Let L1 be the line passing through the points Q1=(−3, 5, −4) and Q2=(−9, −1, 2). Find a value of k so the line L2 passing through the point P1 = P1(−1, 11, k) with direction vector →d=[1, −3, −3]...
1
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3answers
55 views

Ranks of matrix

Find the rank of the following matrix $$\begin{bmatrix}1&-1&2\\2&1&3\end{bmatrix}$$ My approach: The row space exists in $R^3$ and is spanned by two vectors. Since the vectors are ...
0
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0answers
14 views

Are the row vectors in a row reduced echelon matrix always independent?

Are the row vectors in a row reduced echelon matrix always independent? I'm thinking that since the first row is the only row with a non-zero coefficient, then it must be independent of all the ...
0
votes
3answers
11 views

Can’t we use ‘vector product’ to find the angle between two vectors?

There are two vectors : $A = (\hat i + j + k)$ and $B = (\hat i - \hat j - \hat j)$, where $\hat i$, $\hat j$, and $\hat k$ are unit vectors along $x$, $y$, and $z$ axis respectively. We have to find ...
0
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1answer
16 views

Orientation of a normal vector of a plane

I found this question: vector normal to a plane It seems to be related to a problem I'm struggling with, but I need to know what is the rule for the normal vector's orientation. Assuming we want a ...
0
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0answers
12 views

Surface integrals, positive or negative normal?

I'm unsure how to decide whether the normal should be positive or negative in $\hat{n}dS=\pm h_2 h_3 e_1 du_2 du_3$, where $h_i$ are the scale factors, $e_i $ are the base vectors, and $u_i$ are the ...
1
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2answers
46 views

Intuition for orientation of tri-vectors in geometric algebra

I am learning geometric algebra from the MacDonald textbook and it states that the outer product is associative. Letting $\bf{u}$, $\bf{v}$, and $\bf{w}$ be vectors $$\bf{u} \wedge \bf{v} \wedge \bf{...
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votes
3answers
32 views

How do I find an equation of a plane perpendicular to two another planes?

Find an equation of a plane $\pi$ that passes through point $P = (2, 3, −6)$ and is perpendicular to two planes $\pi_{1} : x + y + z − 5 = 0$ and $\pi_{2} : x − y + 2 = 0$. Can someone help me with ...
0
votes
1answer
45 views

Applying the chain rule on vectors and matrices

I need to find $\frac{dy}{dx}$ for the following y = $||A^Tx - b||_2^2$ where $A \in R^{3x3}, b \in R^{3x1}, x \in R^{3x1}, y \in R,$ and $||.||_2$ is the euclidean norm so for example $||z||_2^2 = ...
0
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2answers
20 views

Need some help with analytic geometry

Check if the following point lies on the plane π : 2x − 3y + 4z − 5 = 0 (point) A = (1, -1, 0) Check if the following vector lies on the plane π : 2x − 3y + 4z − 5 = 0 (vector) -AC = (1, 2, 1) ...
2
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2answers
31 views

calculating new 3D position on sphere with angular velocity vector

I feel like this is actually pretty simple but still could not find any solutions so far... I'm trying to calculate the movement of a point in a rigid rod with the equation $ \dot P = [ v + \omega \...
0
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2answers
31 views

Vector understanding (basic)

Im new to vectors and i am confused about the notation. Say you wanted to graphially represent a vector in two dimensional space i get that you can use a directed line segment and we can denote this ...
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1answer
25 views

Vector calculation out of two vectors

When I have vectors $\vec r_1=(4,0,0)$, $\vec v_1=(-2,-1,-2)$, $\vec r_2=(-2,0,3)$, and $\vec v_2=(-1,0,1)$, and unknown vectors $\vec r_3$ and $\vec v_3$ that satisfy $m_1 \vec{r_1} \times \vec ...
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0answers
47 views

Calculate distance between observer and cube excluding the distance insize the cube [closed]

I am trying to calculate the distance between two points where one is an observer and has no size and where the other point is a cube with the dimensions {1, 1, 1}. The distance will be the distance ...
3
votes
2answers
47 views

How does a gradient allow the calculation of the directional derivative

If the gradient only results in a vector telling you the steepest direction to travel, how can the "slope" in any direction be calculated? If the gradient is: $\nabla f(x,y,z) = \left[\begin{array}{...
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votes
1answer
53 views

Find an equation of a plane π that passes through point P = (0, 1, 0) and is parallel to vectors ~a = [−1, 3, 0], ~b = [3, 1, −5] [closed]

I need some help with the second one, can someone help me to solve this?enter image description here
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3answers
24 views

Find the general equation formed by three points

I have to find the general equation defined by these 3 points: What I have tried so far: Unfortunately, I was told that my solution is wrong and I am not sure how else to solve/approach this problem....
0
votes
1answer
23 views

How are the steps to the solution for Arc - Length obtained?

Can someone please help me follow and understand the steps of the solution marked with $(*)$ and $(@)$? Why is the dot product used and computed with the unit vector. How does this equal the integral? ...
1
vote
1answer
15 views

Why should the position vector be noted as $R\hat{R}$ in spherical polar coordinates?

Why should the position vector be noted as $R\hat{R}$ in spherical polar coordinates? Now i did the calculation like this: $\vec R = R \sin\theta \cos\phi \hat{i} + R \sin\theta \sin\phi \hat{j} + R \...
0
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1answer
34 views

Question about vector negation

I have to decide if the following statement is true or false: The operation of vector negation is a bijection from the set of free vectors to itself. Obviously the domain and codomain are correct, ...
0
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0answers
15 views

What is the difference between the following definitions of Vector Functions and Parametric Curves?

The definitions seem exactly the same to me. We have a(t) to be a vector function and x(t) to be a parametric curve where the tip of the position vector traces out a curve in an $n$-dimensional space ...
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0answers
8 views

Cartesian tensors [closed]

would you please help me with these two questions? Given $A_k=\frac{1}{2}\epsilon_{ijk}$ with $B_{ij}=-B_{ji}$, antisymmetric, show that $B_{mn}=\epsilon_{mnk}A_k$. For the three vectors A,B and C, ...
4
votes
1answer
23 views

Linear combination of two vectors in complex space

Let $\mathbf{x},\mathbf{y} \in \mathbb{C}^2$ be two linearly independent vectors in two dimensional complex space. Assume that $\|\mathbf{x}\|\leq \|\mathbf{x} \pm \mathbf{y}\|$. I want to show (or ...