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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Given the point $T$ and the line $p$, find the distance between their orthogonal projections onto the plane $\pi$.

Let $T'$ and $p'$ be the orthogonal projections of the point $T(-8,2,-3)$ and the line $$p\ldots\frac{x}4=\frac{y-4}3=\frac{z+1}{-2}$$ onto the plane $\pi\ldots x-y+3z+8=0$. Find the distance ...
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23 views

Eigenvectors for eigenvalue with multiplicity $\mu = 2$

I'm looking for a way to determine linearly independent eigenvectors if an eigenvalue has a multiplicity of e.g. $2$. I've looked for this online but cannot really seem to find a satisfying answer to ...
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2answers
14 views

Find parametric equations for the midpoint $P$ of the ladder

The following problem appears at MIT OCW Course 18.02 multivariable calculus. The top extremity of a ladder of length $L$ rests against a vertical wall, while the bottom is being pulled away. ...
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3answers
49 views

What exactly does it mean for a vector to have a direction?

Vectors are defined as having magnitude and direction. If I understand it correctly, their magnitude is their length, meaning they have the properties of a line segment. What does it mean for a vector ...
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1answer
30 views

Can a discontinuous vector field be conservative?

I consider a closed curve within a 2-dimensional vector field, passing through two points $A$ and $B$. Going clockwise, along the path from $A$ to $B$ the field is constant $\vec{v}(\vec{x}) = \vec{k}$...
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13 views

Reparameterize the curve $r(t)=\langle e^t\sin t,e^t\cos t,5e^t \rangle$ in terms of the arclength parameter, s with $(0,1,5)$ as the base point.

So first, $r^\prime(t)= \langle e^t\sin t+e^t\cos t, e^t\cos t-e^t\sin t,5e^t \rangle$ Then I took the magnitude of $r^\prime(t)$ which is $\sqrt{(e^t\sin t+e^t\cos t)^2+(e^t\cos t-e^t\sin t)^2+(5e^t)...
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1answer
21 views

Can you take the (Euclidean) norm of a scalar quantity (non-vector)?

Context: This question My question is whether or not you can take the (Euclidean) norm of a scalar non-vector quantity. My intuition is no, but I just wanted to confirm it. I understand that a norm ...
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2answers
101 views

Guidance requested for vector dot product question.

Helping my child out with their year 11 exam preparation, specifically vectors and dot products, I think I may have figured out the answer but I'd like to get some confirmation or, more likely, a ...
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1answer
37 views

tetrahedron $OABC$ of edges $a,b,c$.Let $G_1 $, $G_2 $ be centroids of $ABC $, $AOC $ such that $ OG_1\bot BG_2$, prove $a^2+c^2=3b^2.$ [closed]

In a tetrahedron $OABC$, the edges are of lengths, $|OA|=|BC|=a, |OB|=|AC|=b, |OC|=|AB|=c. $Let $G_1 $ and $G_2 $ be the centroids of the triangle $ABC $ and $AOC $ such that $ OG_1\bot BG_2,$ then ...
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38 views

Find all values for a and b such that two vectors are not perpendicular

I'm stuck on this question that asks to find all values for a and b such that two vectors u = (-1, 2, a-2), and v = (b, 4, -2) are not perpendicular. I know that if the dot product of u.v = 0 then ...
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42 views

Let u and v be two orthonormal vectors in $\mathbb{R}^n$. Then $\| u+v \| $ is [closed]

Select one: a. 2 b. √n c. √2 d. n I'm getting b but i'm confused
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Find the vector function $r(t)$ which satisfies $r'(t)= \langle sin9t, sin9t, 9t \rangle$ and $r(0)= \langle 5,3,2 \rangle$.

I'm not sure how to do this problem at all, but would it make sense to plug in $\langle 5,3,2 \rangle$ into $r'(t)$?
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18 views

Proof of a sustitution on the magnitude of the separation vector.

Given the separation vector and its magnitude: $\displaystyle \vec{r} =( x-x') \ \hat{i} +( y-y') \ \hat{j} +( z-z') \ \hat{k}$ $\displaystyle r=\left[( x-x')^{2} +( y-y')^{2} \ +( z-z')^{2}\right]^{...
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37 views

find the angle between the affine spans

Let $W$ be a vector space and $U,V \subseteq W$ be subspaces. Let $p, q \in W.$ The angle between the affine spaces $P = p+ U$ and $Q = q + V$ (here $p+U := \{p+u: u\in U\}$) is defined as $\theta(P,Q)...
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1answer
26 views

Notation question: dual space basis

I have an exercise that I am trying to decipher, but as I have never seen this notation before I do not know how to read it. The problem states: The vectors $x_1=(1,1,1),x_2=(1,1,-1)$ and $x_3=(1,-...
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2answers
36 views

Points A, B and C have position vectors a = i + 2 j, b = 2i + j and c = 2i − 3 j. Find the shortest distance from B to line AC

Points A, B and C have position vectors a = i + 2 j, b = 2i + j and c = 2i − 3 j. Find: The shortest distance from B to line AC The area of triangle ABC What I have done so far: Found AB and ...
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Dot and cross product in vector algebra

What does dot and cross product mean? Let $a$ and $b$ be two vectors and $\alpha$ be the angle between them. 1)Why do I need to scale up $a$ with the component of $b$ that lies on $a$ (ie. $b\cos(\...
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1answer
25 views

PDF of function of random variable in multidimensional case proof

I read the wiki page on PDF and I got stuck at the proof of relation between pdf $g$ of a function $\textbf{y}$ and pdf $f$ of its random variables $\textbf{x}$: $$g({\bf{y}}) = f({H^{ - 1}}({\bf{y}}...
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17 views

Vectorial solution of a vector equation

Let there be a vector equation $\mathbf A\,+\mathbf B+ \mathbf C=\mathbf0$ Unit vectors of $\mathbf A\,, \mathbf B\,and \,\mathbf C$ are denoted as $A^u\, B^u\,and\, C^u$ respectively and ...
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1answer
32 views

check if vectors are linearly independent?

Heyo, I'm just wondering if I'm correct in assuming that the following three vectors in four dimensional space are linearly dependent.. I simplified them using elemental row operations and ended up ...
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2answers
35 views

Given $a = 3i + 4 j $ and $b = i − j$, find the vector with the same magnitude as $b$ and with the same direction as a.

I understand how to solve this problem: i.e. take the unit vector of $a$, then multiply by our given magnitude (in this case $\sqrt 2$) which gives: $\dfrac{\sqrt 2}{5} (3i + 4j)$. I'm confused as ...
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3answers
30 views

Component vectors from a 3D vector

If vector $v_{xy}$ is a vector on the $xy$ plane of magnitude $r$, and $v_{yz}$ is a vector on the $yz$ plane also of magnitude $r$, then $v_{xy} + v_{yx}$ results in vector $v$ of magnitude $R$. How,...
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2answers
17 views

Let $a$ & $b$ be non-zero vectors such that $a · b = 0$. Use geometric description of scalar product to show that

Let $a$ & $b$ be non-zero vectors such that $a \cdot b = 0$. Use geometric description of scalar product to show that $a$ & $b$ are perpendicular vectors. What I've done so far is to state ...
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1answer
19 views

A particle is moving in $\mathbb{R}^3$ so that its acceleration function is $a(t)=\langle 2t,1,0 \rangle$. FInd the position function, $r(t)…$

A particle is moving in $\mathbb{R}^3$ so that its acceleration function is $a(t)=\langle 2t,1,0 \rangle$. Find the position function, $r(t)$ of the particle if it starts at the point $(-5,0,2)$ with ...
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3answers
90 views

Proving that removing any vector of the linearly dependent set gives a linearly independent set

Consider the matrix representing 6 linearly dependent vectors: $$\left(\begin{array}{llllll} 1 & 0 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 0 & 1 \\ 0 & 0 & 1 ...
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1answer
13 views

Suppose $r(t)$ and $s(t)$ are vector functions. $r(2) = \langle 1,2,−1 \rangle, r'(t) =\langle 3,0,4\rangle$, and $s(t) =\langle t,t^2,t^3 \rangle$.

Suppose $r(t)$ and $s(t)$ are vector functions with $r(2) = \langle 1,2,−1 \rangle$, $r'(t) =\langle 3,0,4\rangle$, and $s(t) =\langle t,t^2,t^3 \rangle$. (a) Find the value of $f'(2)$, when $f(t)=r(...
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1answer
54 views

How to find the limit of a dot product of vectors

For a vector $v=\left( {1 \over 2},{1\over 2},{1\over 2}\right)\;$ the following is defined $$\begin{aligned}w_1&=(e,e+2,e-2)\\ w_n&=v\times w_{n-1}+(2,-4,2)\quad (n\geq2),\end{aligned}$$ $e$ ...
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1answer
46 views

A geodesic on a unit sphere

Points $A(\cos\alpha,0,\sin\alpha)$ and $B(0,\cos\beta,\sin\beta)$, $(0<\alpha$ and $\beta<\pi/2)$ are on a unit sphere and $l$ is the shortest line (geodesic) between $A$ and $B$ on the sphere. ...
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1answer
22 views

unit vector conceptual

For the following, is it always going to be equal to 1? $$\left|\frac{\vec{r}}{\vec{r}}\right|$$
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1answer
37 views

Finding Points On a Circle.

Hello everyone I have 2 points A and B. And I have to proof that for all X points for them $\frac{AX}{BX} = 3$ they are on a one circle. I tried to convert the equation to something like $x^2 + y^2 ...
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0answers
20 views

Projection of vector area onto $x$-$y$ plane

In our math class (in the context of introduction to vectors) we were told that the projection of the vector area $S$ onto the $x$-$y$ plane is equal to the dot product of the vector area $S$ and the ...
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1answer
28 views

Finding the trajectory of a charged particle in space in a magnetic field

I have been given that a charged particle of charge $q$, mass $m$ moving with a velocity $v_{0}(\vec{i}+\vec{j})$ enters a magnetic field $B_{0}\vec{i}$. At any time instant $t$, determine its ...
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1answer
12 views

Show that the linear span of a subset is a subset of the linear span of the superset.

Let V be a vector space over the universal set and S and T be two non-empty finite sets with S $\subset$ T. Show that $<S>$ $\subset$ $<T>$ where denotes the linear span of A. I did ...
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2answers
24 views

How does the linear independence or dependence of the set of column vectors of a matrix depend on that of the set of row vectors of the same?

Let A = ($a_{ij}$) be an mxn matrix. If the set of row vectors of A is linearly independent, is the set column vectors too? What happens if the row vectors are linearly dependent. Does it affect the ...
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0answers
12 views

Can the sides of a special triangle be the components of a unit vector?

So I am wondering if the sides of a special triangle can be used when finding a unit vector and if my approach to this question is appropriate. I will further explain my question below... This is the ...
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1answer
28 views

How to parameterize this following region?

the region which consists of that part of the square $0≤x≤b,0≤y≤b$ lying outside the disk $x^2+y^2≤a^2$. (Take $0<a<b$.)
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27 views

Linear independence between $\cos (x)$ and $\cos(2x)$. [duplicate]

Determine whether $f_1(x) = \cos(x)$ and $f_2(x) = \cos(2x)$ are linearly independent for all real values of $x$. Thank you in advance.
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10 views

Mapping a plane from 3D space to 2D space with non-specific scale

I'm trying to understand how to map a plane in 3-dimensional (x,y,z) space to a 2-dimensional coordinate system. The plane will have ≥3 vectors, but is not necessarily a quadrangle. I have the ...
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2answers
33 views

If vector $c = (a \times b)$, what can be said about $c.a$ and $c.b$? [closed]

I wasn't really sure where to go with this. Would it follow the triple product rule of associativity?
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1answer
25 views

Show that each vector in an n-dimensional vector space can be represented as the summation of its components along the orthonormal basis.

Show that in an n-dimensional vector space V over the universal set with orthogonal basis {$a_1, a_2,..., a_n$}, each vector B can be expressed as: B = $\frac{<B,a_1>a_1}{||a_1||^2}$ + $\frac{...
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0answers
26 views

Verify that a mapping onto a subspace is a projection mapping [closed]

$V$ is a vector space with basis $\{a_1,a_2, \dots a_n\}$ and $W$ is its vector subspace with basis $\{b_1,b_2, \dots,b_k\}$. EDIT - $\{b_1,b_2, \dots,b_k\}$ is a subset of $\{a_1,a_2, \dots a_n\}$....
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0answers
28 views

Find the ratio in which edges are divided in a tetrahedron

In a regular tetrahedron, E and F are the midpoints of edges AD and BC, respectively.M and N are on segments CD and EF, respectively, such that α=∠MNC=π/4, β=∠NME=π/3. In which ratio are EF and CD ...
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1answer
26 views

problem with the vector algebra

I have the vector $\vec c$ that is: $\vec c= \frac{\sum_{i=1}^n m_i\vec r_i}{\sum_{i=1}^n m_i}$ where $\vec r_i$ is a vector and $m_i$ is a scalar I need to proof the folowwing equality for any ...
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2answers
35 views

How to come up with a set of three linearly dependent vectors in a systematic way

Give an example of three linearly dependent vectors in $\mathbb{R}^{3}$ such that none of the three is a multiple of another. Three vectors that satisfy this property are the vectors :$\{(-1,2,1), (3,...
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1answer
40 views

Grade 12 Calculus and Vectors Math Problem

Question: A plane has $-4y + 6z - 4 = 0$ as its Cartesian equation. Determine the Cartesian equation of a plane that is perpendicular to and contain the point $P(-3, -10, 4)$. I tried doing this ...
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1answer
22 views

Find angle BCC¹ on triangular prism

In a triangular prism $ABCA¹B¹C¹$, $|AB|=c$, $|BC|=a$, $|CA|=b$, $\angle BAA¹=\alpha$, $\angle CAA¹=\beta$. Find $\angle BCC¹$ Vector methods are preferred.
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1answer
30 views

Matrix is computed are permuted

what happens to the matrix of a linear transformation on a finite dimensional vector space when the elements of the basis with respect to which the matrix is computed are permuted among themselves?...
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0answers
26 views

Angle between two vectors in nearest degree

Find angle between the vector $\vec{a}=<1,2,2>$ and $\vec{b}=<3,4,0>$ in nearest degree What I try: $$\cos \theta=\frac{\vec{a}\cdot \vec{b}}{|\vec{a}||\vec{b}|}=\frac{3+8}{3\times 5}=\...
2
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1answer
16 views

Symmetric equations of the line intersecting planes

I need help with this exercise. It says: Find the symmetric equation for the line that is defined by the intersection of the planes $x-2y+4z=2$ and $x+y-2z=5$ Using my book as a guide. I see that I ...
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0answers
16 views

Vector function derivation

Let $h_k,v_k\in\mathbb{C}^{N\times1}$ and $C_k,D_k \in \mathbb{R}^{N\times N}$. $K_r$ and $\sigma_{k}^{2}$ are constant. $h_k^H$ denotes the conjugate transpose of $h_k$. A function is formulated as: ...

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