2
votes
1answer
45 views

Curvature kappa with known acceleration, unit normal and unit tangent vectors

The acceleration of a particle is $a(t) = (4\sin t \cos t)T(t) + (4e^t \sin^2 (t/6))N(t)$, where $T(t)$ is the unit tangent vector and $N(t)$ is the unit normal vector. At $t = \pi/2$, the speed of ...
0
votes
1answer
30 views

Vector Cross product - Rearranging issue

Given Data in question I have following relations in vector space$\begin{eqnarray}n_0^{'}(s)=-\kappa(s) \times n_0(s)\\n_1^{'}(s)=-\kappa(s) \times n_1(s)\\n_2^{'}(s)=-\kappa(s) \times ...
0
votes
0answers
34 views

$3D$ surfaces multivariable Calculus

A surface is constructed as follows: First a curve $(0, y, −((y − 1)^2)((y + 1)^2))$ is drawn in the yz–plane. Then a parabola $(u, u^2)$ is drawn in the uv–plane. Finally, in each plane y = b, a copy ...
0
votes
0answers
9 views

Tangential and normal components of acceleration acting on a dropped bomb

The original question is this: A plane flying at an altitude of 34,000 feet at a speed of 510 miles per hour releases a bomb. Find the tangential and normal components of acceleration acting on ...
1
vote
1answer
43 views

Finding a unit vector orthogonal to vectors $a$ and $b$

If I understand correctly, the cross product of vectors $a$ and $b$ is orthogonal to both $a$ and $b$. So for an assignment I have to find two unit vectors orthogonal to vector $a = \langle 1,0,4 ...
1
vote
1answer
20 views

Intersection point between a line and plane: what's wrong with my calculation?

I'm trying to calculate the intersection point between a line and a plane, but apparently there is something wrong with my calculation and I don't know what exactly. The exercise goes as follows: ...
0
votes
1answer
15 views

Intersection point between a line and a plane?

So we have a line, let's called it line L, that passes through (2,−2,1) and (−4,1,−3). We also have a plane, let's call it V, that is given by the equation 3x + 4y + 4z = -42. How can I now ...
1
vote
2answers
59 views

being $\mathbf{w}$ a vector, how do I calculate the derivative of $\mathbf{w}^T\mathbf{w}$?

Let's say that I have a vector $\mathbf{w}$. How can I calculate the derivative in the following expression? $\frac{\mathrm{d}}{\mathrm{d}\mathbf{w}}\mathbf{w}^T\mathbf{w}$ Update: found these ...
0
votes
0answers
27 views

being $\mathbf{a}$ and $\mathbf{b}$ two vectors with same length, how do I expand $(\mathbf{a}^T\mathbf{b})^2$?

Let's say that I have two vectors $\mathbf{a}$ and $\mathbf{b}$. Assuming that they have same length, their product $\mathbf{a}^T\mathbf{b}$ and its square $(\mathbf{a}^T\mathbf{b})^2$ are scalars. ...
0
votes
1answer
23 views

$\nabla \times \underline{v}$ - Results in a vector perpendicular to these two vectors?

Say $v = -y\hat{i} + x\hat{j}$ If we take the cross product of $\underline{v}$ with $\nabla$ we get $\left| \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ \frac{d}{dx} & \frac{d}{dy} ...
2
votes
4answers
92 views

$(1, 1) \cdot (6, 0) = 6?$ Intuition?

$a = (1, 1)$ $b= (6, 0)$ $a \cdot b = (1, 1) \cdot (6, 0) = 6$ I have seen the dot product of $a$ and $b$ refered to as "What is the x-coordinate of $a$, assuming $b$ is the $x$-axis?". Well here ...
3
votes
1answer
53 views

Sums of special vectors

Let $v$ be a vector obtained by taking a sum of $k$ vectors the of the form $(0,0,\ldots,0, -n, *,*,\ldots,*)$, where $"*"$ stands for either $0$ or $1$, and the position of the $-n$ entry can vary ...
1
vote
2answers
293 views

How to proof equality of del dot a cross b

I am trying to prove that $\nabla \dot{}(A\times B) = B\dot{}(\nabla \times A) - A\dot{}(\nabla\times B)$ I tried expanding the RHS but the $x$ component of the vector I am getting is ...
0
votes
2answers
83 views

Definite integral-dot product

I have an integral equation containing dot product $$\int_{0}^{L} \left(\frac{a}{L}.b(s)\right)\mathrm ds\tag 1$$ Data Given a is a constant vector of size 3 b(s) is a varying vector of size 3 " . ...
0
votes
1answer
64 views

Quaternion expansion

I have a quaternion equation $ \psi(s)=Pe^{\frac{1}{2}k(s)}\tag 1$ Given conditions and data Here P is a constant unit Quaternion defined for 3D rotation matrix as $(p_1,p_2,p_3,p_4) , p_4\in ...
1
vote
0answers
31 views

Curve and Constant Curvature

I have initial position vector $p_0$, given curve-linear length $1$. It can be parameterized by $s\in[0,1]$. Assume we have the equation to generate the curve from given starting point and constant ...
-4
votes
2answers
54 views

Orthogonalization of two Vectors [closed]

Given two vectors $v_1$ and $v_2$, which have a given angle $\theta$≠ $$\frac {π}{2}$$, in between; How would one apply a Gram Matrix to define an inner-product, in order to orthogonalize the two ...
1
vote
1answer
23 views

parametric form of vector

I am having trouble understanding what the question is asking at this point, I have solved the first parts correctly and was wondering if I could get help as to how to solve x=x(t)
0
votes
3answers
36 views

magnitude of two vectors

How would I find the crossproduct if all I have is the point values?
0
votes
2answers
28 views

plane which passes through three points

I am confused as to how to answer this question because I don't understand how to incorporate the 12 into my answer. Any suggestions?
0
votes
2answers
349 views

curl of (cross product of two vectors), i know the formula, but not sure how to prove it

$$\text{curl } \left(\textbf{F}\times \textbf{G}\right) = \textbf{F}\text{ div}\textbf{ G}- \textbf{G}\text{ div}\textbf{ F}+ \left(\textbf{G}\cdot \nabla \right)\textbf{F}- \left(\textbf{F}\cdot ...
0
votes
0answers
25 views

Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D)$

As mentioned in the title, my problem is: Show that $\{w^{1/2}\phi_n\}$ is an orthonormal set in $L^2(D)$ if $\{\phi_n\}$ is an orthonormal set in $L^2_w(D).$ So I know that: ...
1
vote
1answer
47 views

How to interperet calculus thing

I have $\nabla \times (f\mathbb{F})$ where $f$ is a twice continuously differentiable scalar field and $\mathbb{F}$ is a twice continuously differentiable vector field. Is it right to interpret $f$ ...
1
vote
1answer
31 views

Can one express $f'(x)$ with the same basis as one uses for $f$?

If I have an orthonormal basis $\{\phi_n\}_1^\infty$ in space $L^2(a,b)$ and the generalized Fourier series expansion for $f$ would be: $$f= \sum \langle f, \phi_n\rangle\phi_n,$$ then can one use ...
0
votes
1answer
34 views

Show that $f(x)$ is orthogonal to $f'(x)$ in $L^2(-\pi, \pi)$

I have the following problem: Suppose $f$ is of class $C^{(1)}$, $\;2\pi$-periodic, and real-valued. Show that $f'$is orthogonal to $f$ in $L^2(-\pi, \pi)$ by a) expanding $f$ in ...
2
votes
1answer
26 views

Proving that $\langle f, g\rangle = \sum_n \langle f, \phi_n \rangle \overline{\langle g, \phi_n \rangle}$

I have the following problem to solve: If the set of functions $\{\phi_n \}_1^\infty$ is an orthonormal basis in $L^2(a,b)$ and the functions $f, g \in L^2(a,b)$, then show that: ...
1
vote
3answers
55 views

Understanding dot product of continuous functions

I'm reading about Fourier analysis and in my book the author speaks about dot product for continuous functions $f, g\in L^2(a,b)$(the set of functions which are square-integrable on the interval ...
0
votes
0answers
48 views

Simple notation questions(2) and unit tangent vector question(1)

I have a vector field $F$, and a rectangle $C$ and some $T$ as a unit tangent vector to $C$ directed anticlockwise around $C$. How is $T$ calculated? Wouldn't it just be a straight line facing ...
0
votes
1answer
31 views

Find the orthogonal projection of the vector… Help appreciated!

Does anyone know how I would go about answering question (b)? (b) Find the orthogonal projection of the vector u = (2,-1) on the vector v = (3,-2). http://i.imgur.com/nWcotnQ.png
0
votes
3answers
41 views

Vectors and orthonormal basis vectors help!

I'm not entirely sure how to go about answering this question about vectors. Any advice/help is appreciated. Write the vector $\displaystyle a =\begin{bmatrix}3\\-1\\7\end{bmatrix}$ as a linear ...
1
vote
0answers
27 views

Jacobian in Change of Variables

Let us consider an integral $\int \mathrm{d} ^ 4 k _ {2} \int \mathrm{d} ^ 4 k _ {1} \, f (k _ {1}, k _ {2})$, where $k _ {1}$ and $k _ {2}$ are four-dimensional vectors in Euclidean space. We want to ...
0
votes
1answer
27 views

evaluating a flux integral

Question: "Region V, of unit volume, is bounded by the closed surface S. Given the vector field $\mathbf{F}=\langle 7x,2y,5z\rangle$, evaluate: $$\int_S \mathbf{F}\cdot\mathbf{dS}$$ I guessed that ...
0
votes
1answer
61 views

Curl and unit vectors after a change of coordinates

Let $x,y,z$ be a system of coordinates with unit vectors respectively $\mathbf{u}_x$, $\mathbf{u}_y$, $\mathbf{u}_z$. Moreover, we have $\nabla = \displaystyle \frac{\partial}{\partial x} ...
0
votes
1answer
29 views

Vectors In Three Dimensions

Hi! I am working on some online homework for my calc2 class and I am having trouble with this problem. I first set $r_1$ and $r_2$ equal to one another to get $(-1-4t, 2+2t, -14+2t)=(-13+4t, 8-2t, ...
0
votes
2answers
50 views

$l^r \subset l^p$ and is it even a subspace

It is true that for $r<p$ and $r,p \in [1,\infty)$ we have that $l^r \subset l^p$. Is it true that $l^r$ cannot be isomorphic to a subspace of $l^p$?
0
votes
2answers
64 views

Multivariable calculus - find total derivative

I want to find the total derivative of the function $f: \mathbb R^n \to \mathbb R^n$, $f(x)=\frac{x}{|x|}$ If I was to copy what the teacher taught, I should find the limit of $\lim_{t \to 0} ...
1
vote
1answer
34 views

Let f: $\mathbb{R}^2 \mapsto \mathbb{R}^2$ be a linear function, proof about $f$ and directional derivative

I have this $f$ that is linear and I want to show that for any $a,v \in \mathbb{R}^2$ $f(\begin{matrix} a_1 + v_1 \\ a_2 + v_2 \\ \end{matrix})$ = $f(\begin{matrix} a_1 \\ a_2 \\ \end{matrix}) + ...
0
votes
1answer
93 views

Proof about non-compact sets and unbounded functions

Let $A \subset \mathbb{R}^n$ be a non-compact subset. Show that there exists a continuous unbounded function on $A$. I have split this into two parts. Either: $A$ is unbounded but closed, or $A$ ...
1
vote
0answers
15 views

How do I prove that group$B(v) = (x : ||x|| \leq 1) $contains an elipsoid around the axes?

I have no idea on where to begin. Also, I have to prove that the aformentioned set is convex, which I tried by contradicting, and I'm having difficulty with that as well. Any help would be ...
1
vote
1answer
43 views

Discrepancy over matrix exponential

I am trying to compute $\large e^A$ for $A = \left( \begin{array}{ccc} 0 & a \\ 0 & 0 \end{array} \right)$ Using $\large e^A = \sum \limits_{k=0}^\infty \frac{1}{k!} A^k$ Writing out the ...
0
votes
0answers
31 views

Tensor calculus

I want to find the following, $\nabla \cdot (\rho u u)$, where $\rho$ is a scalar and $u$ a vector. I get $$ \partial_\alpha(\rho u_\beta u_\alpha) = \rho u_\beta \partial_\alpha u_\alpha + ...
0
votes
1answer
80 views

Find the length and direction of $u \times v$ and $v \times u$

So I was given two vectors: $u=-8i- 2j- 4k$, and $v=2i+2j+k$. I was able to figure out the cross product of $u\times v$ which is $6i-12k$, and $v \times u$ which is $-6i+12k$. However, I need help ...
1
vote
1answer
45 views

Calculus - Check if the line is parallel to the plane

Check if $r = (3,0,2) + t(1,-2,2)$ is parallel to the plane $4x + y - z = 10$ Does it lie in the plane? I'm new to vectors and I'm just wondering how would I solve a question like this!
0
votes
0answers
36 views

Evaluate the surface integral

Let S = $\{(x,y,z) \in R^3$ |$ x^2+y^2+z^2=1\}$.Let V = $(v_1,v_2,v_3)$ be solenoidal vector field on $R^3$. Evalute: $$\int_S [x(x+v_1(x,y,z)) + y(y+v_2(x,y,z)) + z(z+v_3(x,y,z))]dS$$
0
votes
1answer
13 views

Given point A(-4,2,3) and B(4,0,1) what conditions is the line: [x,y,z] = [4,0,1] + t[m,n,1] perpendicular to AB?

Then determine a vector equation either in terms of m or n, of the line that satisfies the condition. Attempt: AB = [8,-2,-2] Therefore, the dot product of [8,-2,-2] and [m,n,1] must be zero. ...
0
votes
1answer
34 views

Solve the linear system in two space?

$(x,y) = (-12,-7) + s(8,-5) $ $(x,y) = (2,-1) + t(3,-2)$ attempt Use elimination: $8s - 3t = 14$ $-5s + 2t = 6$ $s = \frac{10}{33}$ find point: $-12 + 8\cdot\frac{10}{33} = -9.575757$ $-7 - ...
2
votes
2answers
77 views

transforming a vector from cartesian to spherical and cylindrical co-ordinate system

I know the formula(which i don't know how to copy here but it was in matrix form) for transforming a vector from cartesian system to spherical or cylindrical coordinate system. But, I want to know its ...
0
votes
3answers
63 views

Determine the value of k such that the points A(4,-2,6), B(0,1,0), C(1,0,-5) and D(1,k, -2) lie on the same plane.

A(4,-2,6) B(0,1,0) C(1,0,-5) D(1,k, -2) if they lie on the same plane. How can i determine this? How do you know that the points lie on the same plane? Like do i check if they intersect? How ...
0
votes
4answers
3k views

Determine if two straight lines given by parametric equations intersect

Does $[x,y,z] = [4,-3,2] + t[1,8,-3]$ intersect with $[x,y,z] = [1,0,3] + v[4,-5,-9] ?$ Attempt To find out if they intersect or not, should i find if the direction vector are scalar multiples? ...
0
votes
1answer
209 views

How to find $z$-intercept of vector equation

How do I find the $z$-intercept of the vector equation $\left<x,y,z\right> = (6, -2, -3) + t \left<3,-1,-2\right>$ I am so lost, do I set $x$ and $y$ equal to zero, and solve for $z$? I ...