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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21 views

Is there an elegant method to separate a 2-dimensional vector into a diagonal vector and a vector following either the x or y axis?

Hello dear Mathematics users, Is there an elegant method for separating a vector into two vectors, one diagonal and the other collinear to either the x or the y axis, so that the sum of their ...
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1answer
25 views

Expressing polynomial equations as a set of linear equations

Suppose the $4$-vector $c$ gives the coefficients of a cubic polynomial $p(x) = c_1 + c_2x+c_3c^2+c_4c^3$. Express the conditions $p(1) = p(2), p'(1) = p'(2)$ as a set of linear equations of the ...
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1answer
23 views

Integrating unitary vector in calculating stress resultant

The problem defines: velocity in the local base $\{e_r , e_\theta , e_z\}$, the gradient and acceleration in the local curvilignear base $e_{i} \otimes e_{j}$ avec $i, j \in\{r, \theta, z\}$ ...
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28 views

Planet Zog: Gravity of a Sphere with Spherical Deletions

Years ago I was given this problem to do. I couldn't manage it at the time but was given the broad strokes of the solution. I came across it again recently and decided to have a go. Could someone ...
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11 views

Axial vector of a skew symmetric tensor

What is the relationship between the magnitude of axial vector and the magnitude of its corresponding skew tensor? How can we derive it?
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25 views

The exact value of the norm of two vectors

I'm able to find the minimum and maximum values for the norm of the sum of two vectors using the triangle inequality, but I'm not sure if I can find the exact value without knowing the angle between ...
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19 views

Can a flux that varies only in $x$ through surface $ds$ be separated into $dx$ and $dy$ terms

For a vector such as this, if $ds$ was vertical, it could be written as $F(x)dy$. However, as $ds$ varies with $x$, can it be written as a with respect to only $F(x)$, $dx$ and $dy$ terms? The general ...
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54 views

Angle between these two vectors

A mapping tool moves from point $a$ {199,176} to point $b$ {199,164} then moves to point $c$ {198, 185} the movement coordinates in the image shows this. the question is what is the angle that ...
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28 views

$\pi-2\pi$ angle between two 3D vectors

I need to calculate the angle between two 3D vectors. There are plenty of examples available of how to do that but the result is always in the range $0-\pi$. I need a result in the range $\pi-2\pi$. ...
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15 views

Given the orthonormal ordered base pairs $ \hat s_1, \hat s_2$ of $V = R ^ 3 $ below, determine if $\hat s_1, \hat s_2 $ has the same orientation …

Given the orthonormal ordered base pairs $ \hat s_1, \hat s_2$ of $V = R ^ 3 $ below, determine if $\hat s_1, \hat s_2 $ has the same orientation or has opposite orientations. a-) $ \hat s_1$=($\vec ...
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1answer
72 views

What is the physical/phylosophical meaning of geometric (Clifford) product?

This kind of product can hardly be called intuitively understandable. Here is the relevant excerpt of the classical book «New foundations for classical mechanics» by David Hestenes. Hestenes ...
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20 views

Matrix /injective

Let $f: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be given by the matrix $A=\begin{pmatrix} 1 & 1 & 0\\ 0 & -1 & 1\\ 3& 0 & 3 \end{pmatrix}$ 1) I have to determine wheter $f$ ...
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85 views

The non-existence of a matrix $A\in M_3(\mathbb{R})$ such that $A^2-A+I=0$. [duplicate]

I'm trying to get my head around this problem, which states that no $3\times 3$ real-valued matrix can satisfy the equation $$A^2-A+I=0.$$ Of course, the problem is trivial if we consider ...
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Maximization of the sum of the norms of vectors in three dimensions.

Suppose we have a collection of fixed vectors $a,b,c \in \mathbb{R}^3$ and we add another vector $x \in \mathbb{R}^n$ to each of the fixed vectors. How does one maximise the sum of their norms, as in ...
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How to define unit principal normal vector $N(t)=\frac{f '(t)\times(f ''(t)\times f '(t))}{||f'(t)||*||f ''(t)\times f '(t)||}$

Let f(t) be a smooth curve such that $f'(t)\not=0$ for all t. Then we can define unit tangent vector T by $T(t)=\frac{f'(t)}{||f'(t)||}$. So $$T'(t)=\frac{f'(t)\times(f''(t)\times f'(t))}{||f'(t)||^3}$...
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30 views

Is this vector proof question wrong?

If a, b, c and d are not equal or 0 and (a.b)c=(b.c)a`, show that a and b are parallel. Since the dot product is a scalar, I can see that c and a are scalar ...
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1answer
31 views

The identity $||a||^2||b||^2||c||^2 + 2\langle a,b \rangle \langle b,c \rangle \langle c,a\rangle = \sum\limits_{cyc} ||a||^2 \langle b,c\rangle^2$

In the process of solving the problem shown below, I discovered the identity $$||a||^2||b||^2||c||^2 + 2\langle a,b \rangle \langle b,c \rangle \langle c,a\rangle = ||a||^2 \langle b,c\rangle^2 + ||b||...
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1answer
24 views

What does ∇∇:Q mean?

I'm reading a paper from the 1960's on Electromagnetism, and I wanted to know what the symbol ∇∇:Q (where Q is the Electric Quadrupole) means. Is this just the Laplacian? See Below .
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1answer
24 views

How to calculate unique, 3D component vectors from 3D resultant vector knowing the component vector's unique lengths and fixed order?

I'm trying to write a program that can calculate a user's 3D finger pose from only knowing the 3D position of the finger's tip and the lengths of the finger's proximal, middle, & distal phalanx. ...
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1answer
14 views

Write the parametric equations of the line that is perpendicular to the $xz$-plane and contains the point $P(2, -3, 4).$

At first I think I was overthinking it. Since it's perpendicular to the $xz$-plane, I assumed it would be parallel to the $y$-axis, making $m=[0,1,0].$ Which would make the vector equation $$[x,y,z]=...
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39 views

Vector Projection explanation

Please can someone explain why this is? I think I understand projection when it is a comparison of two vectors but below has 3. Im revising and I am so slow. The projection of the vector $$\begin{...
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31 views

Minimum distance between two skewed lines

My doubt is related to minimum distance between skewed lines. Vector AB where b and a are position vectors of AB and points on L1 and L2 respectively. The perpendicular distance PQ is the ...
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45 views

Derivative of a vector-valued function devided by its absolute value.

Let f(t) be a differential curve such that $f(t)\not=0$. Now, how to show that $$\frac{d}{dt}(\frac{f(t)}{||f(t)||}=\frac{f(t)\times (f'(t)\times f(t))}{||f(t)||^3}\tag{1}$$ My attempt: $$\frac{d}{dt}...
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45 views

Solve for the missing vector in a cross product [closed]

How can I answer this question? Solve for all vectors $A$ so that $( 1, 2, 1) \times A = (3, 1, -5)$.
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1answer
16 views

Does only one angle exists between two vectors?

We know that the dot product of two vectors in $\mathbf{R}^3$ is defined as $$ \vec{A} \cdot \vec{B} = a_x b_x +a_y b_y+ a_z b_z $$ Now, if we choose an x-axis such that the vector A lies on it; then, ...
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18 views

Finding torsion of curve $x=\frac{2t+1}{t-1}, y=\frac{t^2}{t-1}, z=t+2$ .

Usually, I construct the radius vector as follows: $$\vec{r} = \frac{2t+1}{t-1}\hat{i}+\frac{t^2}{t-1}\hat{j}+(t+2)\hat{k}$$ Then I find $\frac{d\vec{r}}{dt}$. So, unit tangent $\vec{T} = \frac{d\...
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1answer
24 views

What is the meaning of column-wise gradient and symmetric gradient?

I was studying a literature where in an equation the term $\nabla p_i$ is given and further it is given that $\nabla$ is a column wise gradient and $p_i = p_i(x,t)$, what exactly meaning of column-...
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32 views

Cross section area

A long plank, with a $1 \times 1$ cross section, is cut as shown below. The region of the cut is a parallelogram with sides $\sqrt{2}$ and $\sqrt{3}.$ Find the area of the parallelogram. I recognize ...
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1answer
11 views

Find the directional derivative of a function along the curve.

Question: Find the directional derivative of f=x^2·y·z^3 along the curve x = e^-u; y = 2 sin u + 1; z = u - cos u at the point P where u = 0 My working: At u=0, x=1, y=1, z=-1 so let u = (1,1,-1). ...
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1answer
36 views

fundamental theorem of calculus for vector valued functions

I am studying multivariable calculus and I'm currently doing operations of vector valued functions of the form: 𝐫(t) = f(t)i + g(t)j +h(t)k I came upon a theorem in my textbook that says the ...
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1answer
51 views

What are the necessary and sufficient conditions for a quaternion to rotate one vector so it's pointing in the same direction as another?

This is to solve the following problem : Let $v$, $v'$, $u$ and $u'$ be unit vectors different from each other and built so that there exist a single quaternion that rotates $v$ towards $v'$ that also ...
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1answer
35 views

Existence of linear combination

My book proves the following theorems: Three distinct points $A,B,C$ are collinear if and only if there exist 3 numbers, $\lambda_1,\lambda_2,\lambda_3$, all different from zero, such that $$\...
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1answer
31 views

Angle between the diagonal and edge of a cube (application of dot product)

Question: Find the angle between the diagonal of a cube and one of its edges. I know how to do this question but in particular I do not understand why $\vec b=<1,0,0>$ rather than $<0,0,0>...
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178 views

Make two vectors equal to each other.

Consider you have a array $a=[x_1,x_2,x_3]$ and array $b=[y_1,y_2,y_3]$ and i need to convert $a$ to to $b$ using minimum number of operations. Valid operations are Choose a integer $k$ (by integer ...
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32 views

Diagram shows quadrilateral ABCD, with OA = (-6,3) , OB = (5,5), OC = (7,-2) , OD = (-4,-6). Show that midpoints P, Q, R & S form a parallelogram.

The diagram below shows the quadrilateral ABCD, with OA = (-6,3) , OB = (5,5), OC = (7,-2) , OD = (-4,-6). Show that the midpoints P, Q, R & S form a parallelogram. My working out so far: ...
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1answer
39 views

Why is the derivative of the binormal vector parallel to the normal vector?

In my notes, it says to consider an arc length parameterization r(s). Then we can show that B' points in the direction of N such that we can write B'=-𝜏N. I understand why ||B'||=𝜏 but am unsure how ...
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1answer
23 views

How to solve $k_b \bar b - k_a \bar a + \frac{1}{t} \cdot k_a \bar a - \frac{1}{t}(\bar c - \bar a_1) = \bar a_1 - \bar b_1$

I need to solve $k_b$, $k_a$ and $t$ in this equation in a program I'm creating: $$k_b \bar b - k_a \bar a + \frac{1}{t} \cdot k_a \bar a - \frac{1}{t}(\bar c - \bar a_1) = \bar a_1 - \bar b_1$$ All ...
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2answers
67 views

How to find linearly independent vectors?

There are three vectors: $$a_1 = (-1, 1, 0, x)\\ a_2 = (2, -3, 1, 2)\\ a_3 = (1, -2, 1, -1)$$ How can I find the parameter x so that these vectors are linearly independent? I'm not quite sure how to ...
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34 views

Prove by using vectors $(\vec {AP} +\vec {CP}+\vec {BP})=\vec 0$ [closed]

Let A,B,C be the points of triangle and P inside of triangle. How can I prove using vectors that: $$(\vec {AP} +\vec {CP}+\vec {BP})=\vec 0$$ only if P is centroid
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1answer
31 views

Applying span twice to a set of vectors

Consider a line $\ell$ in $\mathbb R^2$ that passes through the origin. What happens when one applies span twice? Particularly, what is $\operatorname{span} \{\operatorname{span} \{\ell \} \}?$
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1answer
15 views

What is the relation between the direction cosines of a vector in an nonorthogonal coordinate system?

If an orthogonal coordinate system transforms to another system of coordinates which is not orthogonal, then does the relation between the direction cosines in the orthogonal system, i.e. $l^2+m^2+n^...
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1answer
33 views

How are velocity vectors calculated when speed and change vector is given?

This is related on an insanely hard geometry problem on Codeforces that I am struggling with: RC Kaboom Show The jist of the problem is as follows: Given an infinite 2D field, you can place $n$ ...
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1answer
23 views

Condition for collinearity of points?

Under which of the following conditions will the points $A, B, C$ with position vectors $\vec a$, $\vec b$ and $\vec c$ respectively be collinear? (a) $\vec c-\vec a = 2(\vec b-\vec a)$ (b) $|\vec ...
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33 views

Gradient of an angle

We have three points labeled 1, 2 and 3 and an angle in the following image. It is easy to show that the gradient of $\theta$ with respect to the position of the point 1 (i.e. when the other two ...
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25 views

What is the intuition behind a Frobenius norm?

I am reading up on how to find the closest correlation matrix and they approach it by minimizing a weighted Frobenius norm. Now I am trying to understand the intuition as to why they use this. Let's ...
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1answer
26 views

Find a vector from this equation

I have a directional vector $L_d$, that is the last direction that something has been moved. This vector is normalized. I also have the target point $T$ of this directional vector. The problem is, ...
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1answer
19 views

What is the explanation about the difference in two derivatives of vectors?

My attempt: a) $g(t)=t^3*\vec{c}$ curve is parallel to $\vec{c}$ because $\vec{c}\times t^3*\vec{c}=[0,0,0]$. b)$h(t)=e^t*\vec{c}$ curve is also parallel to $\vec{c}$ because $\vec{c}\times e^t*\vec{...
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1answer
20 views

What is the probability that a uniformly random vector is spanned by a subset of basis vectors?

Working in the set of Real Numbers. Say that you're given a set of basis vectors $\{\mathbf{b}_i\}, \, i \in [n]$. Now, sample a vector v uniformly at random from the entire space. What is the ...
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2answers
54 views

finding locus of a point in 3d.

Find the locus of the point which moves so that its distance from the line x=y=z is twice its distance from the plane x + y+z=1. I know the distance of point (x,y,z) from given plane will be mod(x+...
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2answers
31 views

Angle between a line on a plane and its projection on to a different plans

Two planes π1 and π2 intersect in a line l. The angle between planes π1 and π2 is 45◦.Let A be a point on l, and let m be the line in plane π1 passing through A that is perpendicular to l. Let B be a ...