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

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1answer
26 views

Unit outward normals

Consider the following region $\Omega_a \subset \mathbb{R}^2$ for $a \in \mathbb{R}$: $$\Omega_a=\{(x,y) \in \mathbb{R}^2 \mid y \ge x^a, x \not= 0 \text{ if }a < 0\}$$ Let $\nu_a$ denote the unit ...
0
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2answers
37 views

What result multiplying 2 3D-vectors? [on hold]

I understood that multiplying two vectors by cross multiplication(!) results in a third vector which is orthogonal to the two first. What does multiplying 2 3D-vectors give us as a result? What ...
0
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1answer
20 views

Length of projection onto a subspace equal length of the vector

If the length of the projection of a vector onto a subspace equals the length of the vector, does this always imply that the vector belongs to that subspace. This was quite easy to show when the ...
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3answers
28 views

Scalar form of magnitude of vector quadratic

$\def\b#1{\mathbf#1}$ I know that $\|\b{b}t + \b{c}\|$ can be written as $\sqrt{ (\b{b} \cdot \b{b}) t^2 + 2(\b{b} \cdot \b{c})t + \b{c} \cdot \b{c}}$ Is there a similar expression for $\|\b{a}t^2 + ...
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1answer
17 views

Find the equation of the plane that goes through the origin and is parallel to the lines

Find the equation of the plane that goes through the origin and is parallel to the lines $$\mathbf{R_1}=3\boldsymbol{\hat{\imath}}+3\boldsymbol{\hat{\jmath}}-\boldsymbol{\hat k} +s(\boldsymbol{\hat ...
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0answers
7 views

Discretization of Unit Vector in 3D

I cant think of a thing that I think is supposed to be easy... =/ Im glad if you could help me. Im working with a regular discretization of a 3d euclidean space. Cubic cells. Then, after a ...
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2answers
18 views

How to find the space vectors and the dimension for this subspace?

It is given that $V=\Bbb {R}^{2 \times 3}$ and $U= \{ B=(b_{ij}) \in V \mid b_{11} + b_{12} + b_{13} = 2(b_{21} + b_{22} + b_{23}) \}$ is a subspace of $V$. I need to find the space vectors of $U$ ...
0
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1answer
24 views

How can I derive the resultant of 2 bearing/elevation pairs

Say, for example I have a gimballed camera mounted on a metal plate, which is itself fixed horizontally to a boat. I can measure the elevation and bearing of both the camera with respect to the plate ...
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0answers
18 views

What would be line integral along path number (iv) [on hold]

In the above image what should be the line integration along path iv. Thanks.
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1answer
23 views

Find the equation of the plane - intersects [on hold]

Find the equations of the planes, which intersect in the line $x-1=2-y=(z-3)/2$. Describe in words how you found the planes. I'm stuck on this question. Help would be very much appreciated
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2answers
33 views

Notation for a unit vector

If I have a vector $\vec{v}$, is there a standard concise notation for the unit vector in the same direction of $\vec{v}$ that is $\frac{\vec{v}}{|\vec{v}|}$?
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3answers
42 views

Find the point of intersection of two lines [on hold]

Find the point of intersection of two lines $l_1: x=2t+1, \quad y=2+3t, \quad z=4t+3$ $l_2: x=s+2, \quad y=2s+4, \quad z=-4s-1$ And find the Cartesian equation of the plane that ...
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3answers
25 views

Vector planes - equation of the plane

What is the equation of the plane that passes through the points $A(1,2,3)$ and $B(3,2,1)$ and is orthogonal to the plane $4x-y+2z=7$? I do not know how to start. Am I searching for the intersection ...
0
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1answer
34 views

cross product of vectors - distance from a line.

I have the following question : Let $A$ and $B$ be distinct points in $\mathbb R^3$. Show that the distance, d, of the point $P\in{\mathbb R^3}$, from the line through A and B is given by ...
2
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1answer
29 views

Demonstrate that the two formulas for a scalar product are equivalent.

In the figure below, three vectors are joined together to form a triangle. The name of each vector is a single letter in boldface, each vector is specified by three lengths in an $xyz$ coordinate ...
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0answers
29 views

Vector Identities [on hold]

I am trying to simplify a vector equation, and I feel like this is something, but I am not sure what. Has anybody seen something like this before? $\|\mathbf{a}\|^2 ...
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2answers
50 views

If a = 3i + 2j and b = -7i + 4j, find a + b as…

"Trig functions enable you to make mathematical models of vector quantities:" If $\vec{a}$ = 3$\vec{i}$ + 2$\vec{j}$ and $\vec{b}$ = -7$\vec{i}$ + 4$\vec{j}$, find a + b as: A) a sum of two ...
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2answers
45 views

Closest to Another Vector Within a Given Angle

Given: 3 component vectors: $\vec x$ and $\vec y$ Angle $\theta$ The angle between $\vec x$ and $\vec y$ is greater than $\theta$ Find a 3 component vector $\vec z$ such that $\vec z$ is in the ...
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1answer
29 views

Prooving Kepler's Second Law through vectors.

I am taking a multivariable calculus lecture online provided by MIT OpenCourseWare. ...
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1answer
23 views

An application of Greens's theorem

Apply Green's theorem to prove that, if $V$ and $V'$ be solutions of Laplace's equation such that $V=V'$ at all points of the closed surface $S$, then $V=V'$ throughout the interior of $S$. ...
0
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1answer
14 views

Vectors and Cartesian Equation

Consider a plane with the x,y, and z-intercepts equal to a, b, and c respectively. Find the Cartesian equation of the plane. I know that the Cartesian of a plane is Ax + By + Cz + D = 0, but I'm not ...
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1answer
14 views

Compute flux of vector field F through hemisphere

I need help solving this question from my textbook. Compute the flux of the vector field: $$\vec F = 4xz\vec i + 2 y\vec k$$ through the surface $S$, which is the hemisphere: $x^2 + y^2 + z^2 = 9 , ...
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0answers
14 views

An electron in a TV tube is beamed horizontally at a speed of 4.3 * 10^6 meters per second

An electron in a TV tube is beamed horizontally at a speed of 4.3 * 0^6 meters per second toward the face of the tube 31 cm away. How far will the electron drop before it hits? (Assume ideal ...
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1answer
22 views

how to rotate scaled-vector (orientation) by scaled-vector (rotation)

Recently I got the physics-engine portion of my 3D simulation / game engine working correctly. The most convenient way to store and compute position and orientation are in 3-element vectors (though my ...
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2answers
45 views

How to solve this vector equation with magnitude constraint?

$\def\b#1{\mathbf#1}$ Here's the equation where $ \b{a}$ and $t$ are unknown (all boldface variables are $3$ element vectors and $t$ is a scalar): $$ \tfrac12 \b{a}t^2 + \b{v_0}t + \b{p_0} = ...
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1answer
19 views

Find a position vector $r(t)$ at time $t$ (word problem).

A particle traveling in a straight line is located at point $(9,8,-9)$ and has speed $5$ at time $t=0$. The particle moves toward the point $(4,6,6)$ with constant acceleration $(-5,-2,15)$. Find its ...
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0answers
6 views

What force should be applied to each axis given a forward angle and a left/right direction?

I have three parameters: up/down angle, left/right angle, and power. The up/down angle can range from 0-90. It determines the forward force applied to my object. 0 means it moves straight forward ...
1
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1answer
28 views

Linear algebra does the given line intersect plane?

Determine whether the line $x = (-1, 0, 1) + t(1, 2, 4)$ intersects the plane $2x-y+z=5$. Find the point of intersection if they intersect. I know the equation follows the form $x = p + td$, so I ...
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2answers
59 views

Find the point on the line $y=2x+1$ that is closest to the point $(5,2)$.

I already attempted this problem this problem with the formula for Vector projection $(x^\text{T}y)/(y^\text{T}y) y$ and go the solution $(1.1,3.3)^\text{T}$ but the book states the solution is ...
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1answer
18 views

What is the correct way of writing a vector element?

When using the convention making a label bold to indicate a vector, should you still use the bold if you are only referring to a single element of the vector? for example, which of these methods of ...
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3answers
39 views

Find all $2 \times 2$ matrices ${A}$ that have the property that for any $2 \times 2$ matrix ${B}$, ${A} {B} = {B}{A}.$ [duplicate]

Find all $2 \times 2$ matrices ${A}$ that have the property that for any $2 \times 2$ matrix ${B}$, ${A} {B} = {B}{A}.$ A hint says that the given equation must hold for all ${B}$. Try matrices ${B}$ ...
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6answers
1k views

“Vectors aren't really numbers” - how sound is that statement?

Since I first learned about vectors, I noticed something interesting: almost any numeric formula can be replaced by a vectorial formula by just replacing addition, multiplication, etc., with their ...
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1answer
13 views

*scVector Differential Equation

What's the general method for solving differential equations of this type $$\frac d{dt} \pmatrix {y \\ x} = \pmatrix{y_0 \\ x_0} + y\pmatrix{a \\ b}$$ where $y=y(t), x=x(t)$, and $y_0$, $x_0$, $a$, ...
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2answers
31 views

How to find direction and normal vector?

This might seem strange, but I can't really understand how to get direction vector for a given edge. For example: $6(x+10)=7(y+20)=7z$, and the direction vector should be $(7,6,6)$. Another example: ...
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3answers
22 views

How do I find a dual basis given the following basis?

$V = \Bbb{R}^3$ and has basis $\mathcal{B} = \{\vec{e_1}-\vec{e_2},\vec{e_1}+\vec{e_2},\vec{e_3}\}$ How do I find the dual basis? This is not homework, but an example that I am struggling to grasp. ...
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3answers
36 views

Find the 2x2 matrix given 2 equations.

Find the $2 \times 2$ matrix ${A}$ such that ${A}^2 = {A}$ and ${A} \begin{pmatrix} 7 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}.$ I have tried to express A as a matrix with variables ...
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1answer
23 views

Vector calculus problem, constant speed, counterclockwise or clockwise.

I'm stuck on how to do this problem: $\displaystyle \vec{r}(t)=(\cos t)\,\vec{i}+(\sin t)\,\vec{j}, \qquad t \geq 0.$. Does the particle have constant speed? (yes or no) For this one I was thinking ...
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0answers
18 views

Vectors Core question [closed]

I have a guestion wich I do not understand or just do not know how to get to the right answer ABCDEF IS A REGULAR HEXAGON IN WICH VECTOR BC REPRESENTS b AND VECTOR FC represents 2a Express the ...
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0answers
10 views

Mean Squared Error in PCA [closed]

Consider a $m$ dimensional data vector $x_i$ which is projected along a unit vector $v_i$. How to calculate mean square error between $x_i$ and its projection vector? What would be the generalized ...
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2answers
29 views

How to prove that $(\vec n\cdot\nabla)\vec n$ is orthogonal to $\vec n$ for unit vector $\vec n$?

I'm trying to prove or disprove that if $\vec n(x,y,z)$ is a unit vector, then $(\vec n\cdot\nabla)\vec n$ is orthogonal to $\vec n$. For this I first tried to compute $\vec n\cdot((\vec ...
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0answers
20 views

Sketching the vector $u - v + w$ using vector addition and subtraction?

I am currently trying to do vector addition and subtraction and I am wondering if what I am doing is correct in any way? So given, the graph with the three vectors drawn out, we have these following ...
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2answers
40 views

What is the difference between $n$-tuples, $m \times 1$ and $1 \times n$ matrices?

Isn't the tuple different structure from $m \times 1$ or $1 \times n$ matrix? Why can you mix them?
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0answers
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Find a vector equation for the tangent line to the curve r⃗(t)=(3cos(2t))i⃗+(3sin(2t))j⃗+(sin(3t))k⃗ at t=0. [closed]

So I was thinking of finding the derivative of this vector, and I got <-6sin(2t), 6cos(2t),3cos(3t)>. I plugged in 0 for t and I got <0,6,3>, but that is not the correct answer since i have to ...
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1answer
25 views

How would I find the acceleration of this vector?

So I found the velocity already (which my homework says it's correct). The velocity is i+2tj+4k. I know the acceleration is the derivative of the velocity. I found it to be 1+2j+4 , but when I enter ...
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1answer
53 views

Find a function $f(x)$ so that the graph of $y=f(x)$ is the path of the particle.

The equation $r(t) = \frac{t}{t+4} \vec i + \frac{4}{t} \vec j$ gives the position of a particle in the $xy$-plane at time $t$. Find a function $f(x)$ so that the graph of $y=f(x)$ is the path of ...
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1answer
28 views

Distance between parallel vectors

Compute the distance between the parallel lines given by $\begin{pmatrix} 1 \\ 4 \end{pmatrix} + t \begin{pmatrix} 4 \\ 3 \end{pmatrix}$ and $\begin{pmatrix} -5 \\ 6 \end{pmatrix} + s \begin{pmatrix} ...
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0answers
26 views

What is the intersection of a plane and a trajectory? [closed]

It's a generic question as I really struggle with vectors and have found nothing of any help on the internet. I'm guessing it has something to do with making the 2 equal to each other and solving ...
0
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2answers
72 views

What are the “building blocks” of a vector?

Lets say I have a set of vectors $V$ that includes this vector: $$\begin{bmatrix}1\\2\\-1\end{bmatrix}$$ I interpret it as $x = 1, y = 2, z = -1$ (that being three dimensions for this vector). I know ...
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0answers
16 views

Calculate new pitch and roll after rotating about the z axis

I am wanting to know how to find out the new pitch and roll values when rotating around a circle. I have become a little stuck on how to achieve this, but hopefully someone will be able to point me in ...
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1answer
72 views

What does this vector notation really mean?

With regard to vectors, how is this (form 1): $$\begin{bmatrix}1\\2\\-1\end{bmatrix}$$ Different to this (form 2): $$\begin{bmatrix}1\ 2\ -1 \end{bmatrix}$$ I would think that the first set consists ...