# 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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### Finding vertical projection of $\vec{a}$ on $\vec{b}$

I want to find the vertical or you could say perpendicular component of $\vec{a}$ on $\vec{b}$ Now I know that it can be found out using $\vec{a} - \left(\frac{a\cdot b}{|b|}\right)\vec{b}$ However I ...
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### Proof of Vector Triple Product by Directions and Magnitudes

I'm trying to prove the vector triple product expansion by magnitude and direction: $$\vec{a} \times(\vec{b}\times \vec{c})=(\vec a\cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c}$$ The ...
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### Find the resultant force

enter image description hereFind the resultant force when these three forces are applied to point A. I have tried to combine Force 1 and Force 2 together first, and used the resultant force to combine ...
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### Geometrical explanation / visualisation as to why 2 vectors cannot span R3?

I have already understood as to why 2 vectors cannot span R3 in a simple algebraic manner. This is from an answer to this very question 3 years ago : Why two vectors cannot span ${\bf R}^3$? However, ...
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### To find the angle a particle makes with the horizontal at any time 't'

Should you vector sum the position vectors at this time 't' or vector sum the velocity vector at this time 't' to find the angle a particle makes with the horizontal at any time 't'
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### Vector methods to find line of intersection of two planes

If we are given two non-parallel planes $A$ and $B$ where $\textbf{a}$ and $\textbf{b}$ are unit normal vectors to $A$ and $B$ respectively, and we are also given that the points with position vectors ...
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### Let a,b,c be distinct non-negative numbers. If the vectors ai+aj+ck,i+k and ci+cj+bk lie in a plane, then relation between a, b and c

My Question is How to proceed with the question. I mean there are two ways:- Either one could apply the scalar triple product and solve the determinant equated to 0 which has provided me with the ...
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### Let P and Q are two points on curve $y=\log_{1/2}(x-\frac 12) + \log_2(\sqrt{4x^2 -4x +1})$ and P is also on $x^2+y^2=10$..

Let P and Q are two points on curve $y=\log_{1/2}(x-\frac 12) + \log_2(\sqrt{4x^2 -4x +1})$ and P is also on $x^2+y^2=10$. Q lies in the given circle such that abscissa is an integer. Find the ...
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### What is a Surface

Let $f$ be a differentiable function and $c$ a number, The set of points $P$ s.t $f(P)=c$ and $\nabla f(P)\ne 0$, is called a surface. (The author didn’t give any informations about the domain and the ...
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### Transformation matrix for rotation by angle theta [closed]

Find the standard matrix of the transformation T: R3 → R3 that corresponds to the anti-clockwise rotation by an angle θ about the x1-axis. I have no idea how to even begin to approach this problem. ...
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### Vectors and polar coordinates [closed]

First part Second Part I understand parts i and ii. I am confused on parts 3 through 6.
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### Matrix norm ineequality

Let $A= \begin{bmatrix} a & b \\ c & -a \end{bmatrix}$ with $a, b, c ~$ are complex nubers satisfying $0\leq a <1$ and $\vert b \vert=1$. Prouve that $$a^{2}+1\leq \Vert A \Vert .$$ I ...
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### i have a set of vectors and i have to check if the sentence is true or not [closed]

i have a question that goes this way i am given S=(v1,v2,v3) and S is group of 3 vectors From vector space V and this question is if S is Linear dependent then v3 belongs to span{v1,v2} if dim(V)=3 ...
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### Prove $P^k = P$ for all $k=1,2,\ldots$ [closed]

Let $P = A(A^TA)^{-1}A^T$, where $A$ is an $m \times n$ matrix of rank $n$. Prove that $P^k = P$ for all $k=1,2,\ldots$
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### Orthonormalized basis of linearly dependent system

Given the following vectors: How is the orthonormalised basis of the span{v1, v2, v3} calculated? Given it is linearly dependent do you just need to work out the orthonormalised basis of the span {v1,...
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### Proving that $\Vert Ax\Vert \leq \Vert A\Vert \Vert x\Vert$ where $A$ is a matrix and $x$ a vector

I would like to prove that $\Vert Ax\Vert \leq \Vert A\Vert \Vert x\Vert$ where $A$ is an $n\times n$-matrix and $x = \left(\begin{array}{c} x_1\\ \vdots\\ x_n\end{array}\right)$ is a vector. I've ...
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### how can I prove $(A\cdot\nabla)A=\frac12\nabla A^2-A×(\nabla×A)$? [closed]

$$(A\cdot\nabla)A=\frac12\nabla A^2-A×(\nabla×A)$$ help me prove this So i try to proof this equation. We're talking about vectors, divergence, and curl. But i dont really know how come the left side ...
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### Shortest distance between two particles whose position vectors depend on time

The Question At time $t$, two particles have position vectors given by r$_1(t)$ and r$_2(t)$ where $$\mathbf{r}_1=\mathbf{a}_1+\mathbf{b}_1\left(t-t_1\right)$$ \mathbf{r}_2=\mathbf{a}_2+\mathbf{b}_2\...
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### plotting 2 x 2 matrix visually. What it represents?

Hello I am engineering student. I need help understanding basic of matrix plotting. Currently I am learning about vectors and I wonder how they are represented by a matrix. Suppose we have a 2D vector ...
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### Why is the term “$1 \times k$ vector” used rather than $k$-length vector?

I see well informed people using the term "$1 \times k$ vector" in lectures. My understanding is that a vector is necessarily $1 \times k$, such that the "$1$ times" part is ...
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### How can I analyse a vector transformed by a special symmetric matrix?

Given a vector $\mathbf{x} \in \mathbb{R}^{N}$, a matrix $\mathbf{A}\in \mathbb{R}^{M\times N} (M\ll N)$, and a constant positive real number $\rho$, my target vector $\mathbf{y}\in \mathbb{R}^{N}$ is ...
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### Finding the tangent to a parametric curve $(t^3, t^5)$ at $(0,0)$

The curve $(t^3, t^5)$ at that point $(0,0)$ does not have a tangent vector as when you work it out, you will arrive at $(0,0)$. Question: How can you find a new parametrisation for the curve such ...
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Let $\mathbf{c}$ and $\mathbf{d}$ be fixed vectors in $\mathbb{R^3}$. Find all vectors $\textbf r$ such that $\textbf r+\textbf r\times\textbf d=\textbf c$. Attempt: Take the vector product with $\... 1answer 47 views ### The angle between two vectors I'm just a new person here and hope that I can ask my question properly. Here is my question: Find the vector$\vec{b}$, when the angle between two vectors is$30^\circ$and one vector is given by$\...
A spaceship is traveling through space at constant speed along a straight line that passes through the points $A = (-3,-8,-6)$ and $B = (-8,-4,-3)$. The star Gliese $061$, located at $P = (-12,0,0)$ ...