Vector algebra Definition and 75 Threads

My book says, If the position vectors a, b, c of three points A,B,C and the scalars α, β, γ are such that αa + βb + γc=0,and α + β + γ = 0, then the three points A,B,C are collinear. On the other hand, If the position vectors a, b, c, d of the four points A,B,C,D (no three of which are...

Hi, a doubt about the definition of vector space. Take for instance the set of polynomals defined on a field ##\mathbb R ## or ##\mathbb C##. One can define the sum of them and the product for a scalar, and check the axioms of vector space are actually fullfilled. Now the point is: if one...

Heres how I proceeded, Equation of line ##AC## in vector form: $$\vec r=a+t(c-a)$$$$\vec r=(1i+4j+3k)+t(2i-6j+2k)$$ Since ##B## doesn't lie on ##AC## ##b\neq (1+2t)i +(4-6t)j+(3+2t)k## The following equation is derived: $$2\hat i+\alpha \hat j+4\hat k\neq (1+2t)\hat i +(4-6t)\hat j+(3+2t)\hat k...

I am extremely confused with how to represent vectors that do not start at the origin in spherical coordinate system. If they did start at the origin, the vector to any point is simply ##r\pmb{\hat{r}}##; however, what if it does not start at the origin as in the question above? One thing I can...

In the following%3A%20https://pubs.rsc.org/en/content/articlehtml/2013/sm/c3sm00140g?casa_token=3O_jwMdswQQAAAAA%3AaSRtvg3XUHSnUwFKEDo01etmudxmMm8lcU4dIUSkJ52Hzitv2c_RSQJYsoHE1Bm2ubZ3sdt6mq5S-w'] paper, the surface velocity for a moving, spherical particle is given as (eq 1)...

I'm not sure where to start, when I tired using integration of the initial equation to get pos(t)=-.65t^2 i + .13t^2 j + 14ti +13tj but after separating each component, i and j, and setting j equal to zero I got 0 or -100 seconds which doesn't seem like a reasonable answer.

The question I have is that if the aero plane is traveling in the same direction as the wind, would it not increase its velocity, as in with boats and streams? So, if by chance, ##w = v##, then the velocity of the aero plane would double. It feels weird as going by the same logic, if the speed...

Hello, I am trying to solve a problem and I would like to ask for help. I have 3 points (A, B, C) in 3D space that are assumed to be on a circle. EXAMPLE 1 EXAMPLE 2 My goal is to create an algebraic formula to calculate the coordinates for 10 points on a circle composed of ABC points at...

First of all, sorry for the title I don't know the name of this formula and that's part of the problem, I can't find anything on google. I have to show the identity above. Here's what I did. I don't know if this is correct so far. ##\vec{u} + \vec{r}(\vec{\nabla} \cdot \vec{u}) + i(\vec{L}...

Hello, This question is with regards to the discussion around page 56 (1971 Edition) in Anthony French's Newtonian Mechanics. He is discussing the choice of a coordinate system where the axes are not necessarily perpendicular to each other. Here is the summary of what I read (as applied to...

In the below figure how triangle OAP is right angled. I have imagined everything but I cannot imagine angle A as right angled. Thanks!

My attempt is below. Could somebody please check if everything is correct? Thanks in advance!

How I would have guessed you were supposed to solve it: What you are supposed to do is just take the gradients of all the u:s and divide by the absolute value of the gradient? But what formula is that why is the way I did not the correct way to do it? Thanks in advance!

Can anyone please help me see if my reasoning is correct regarding the following question? I'll just solve for the case where the dinghy tracks so as to just 'touch' the exclusion zone on the 'high' side So, in the diagram below: The dinghy tracks along the red path, inclined at x degrees...

Could I please ask for help regarding the last part of this question: At a given instant, a ship P revelling due east at a speed of 30km/h is 7km due north of a second ship Q which is traveling x degrees west of north at a speed of 14km/h, where tan(x)=3/4. Show that the speed of Q relative to...

Hi all, I can't find a single thing online that translates a cartesian velocity vector directly to spherical vector coordinate system. If I am given a cartesian point in space with a cartesian vector velocity and I want to convert it straight to spherical coordinates without the extra steps of...

Summary:: the set of arrays of real numbers (a11, a21, a12, a22), addition and scalar multiplication defined by ; determine whether the set is a vector space; associative law Question: determine whether the set is a vector space. The answer in the solution books I found online says that...

Okay, so the answer is quite easy if you draw a diagram and notice that cosine law solves everything rapidly. But at first, I tried doing some vector algebra and apply properties to see if I could get to something. This is what I could develop. Consider ##|\vec u|##=12, then $$\langle \vec...

We all know that the area of a triangle having consecutive sides as ##\vec { a }## and ##\vec { b }## has the area ##\frac { 1 } { 2 } | \vec { a } \times \vec { b } |## but what is the direction of that area vector? I mean if we consider ##\vec { a } \times \vec { b }## that will be one...

Homework Statement Consider the real-vector space of polynomials (i.e. real coefficients) ##f(x)## of at most degree ##3##, let's call that space ##X##. And consider the real-vector space of polynomials (i.e. real coefficients) of at most degree ##2##, call that ##Y##. And consider the linear...

Homework Statement Three vectors are given: A=2i+3j, B=1i+5j, C=-1i+3j Find constants x and y such that xA+yB=C Homework Equations N/A The Attempt at a Solution The form of the final equation reminded me if standard form of a slope, so I found the total vector for A,B, and C. I was then going...

Homework Statement This is a solved problem, but I haven't understood a few things. I've marked out sections of the solution in white for convenience. The markings are positioned where that particular section ends. In part (1), how did they just assume f1(0) = 2, f2(0) = 3, g1(0) = 3, g2(0)...

Homework Statement A river of width 4 km flows at 5 km/h. A swimmer whose speed relative to water is 4 km/h, starts swimming from a point A on a bank. What minimum distance will the swimmer have to walk on the other bank to reach point B directly opposite to A? [/B]Homework Equations Vba = Vb...

Homework Statement This is from Griffith's Introduction to Electrodynamics, where the book is deriving the magnetic dipole moment from multipole expansion of the vector potential The vector potential of a current loop can be written as $$\mathbf{A(r)}=\frac{\mu_0 I}{4\pi} \left[ \frac{1}{r}...

Homework Statement |a| = 2 |b| = ## \sqrt3## |a - 2b| = 2 Angle between a and bHomework EquationsThe Attempt at a Solution ##\theta## is angle between a and b So angle between a and -2b is 180-##\theta## [/B] ##|a-2b|^2## = |a|^2 + |2b|^2 -2|a||2b|cos(180-##\theta##) ##2^2## = 2^2 +...

1. Problem A boatman crosses a river of width ##D## from a point ##O##, looking to get to point ##A## on the opposite riverbank. Suppose that the flowrate is uniform with velocity of magnitude ##v_0##. The boat has a velocity ##\vec{v_1}## relative to the water, with constant magnitude, and it...

My question is simply whether you can reduce a vector triple product, or more generally a scalar multiplier of a vector in a cross product? Given: (A x (uB x C) = v, where u and v are known constants. Is it valid to change that to: u(A x (B x C) = v or (A x uB) = v, can you change that to u(A...

Homework Statement Homework EquationsThe Attempt at a Solution I took ## \frac { d\vec r}{dt} = 0 ## . This gives ##t_m = \frac { 1} {\sqrt{ 2}} ## ...(1) To calculate ## \alpha ## , ## \cos{ \alpha } = \frac { \vec v \cdot \vec r}{|\vec v||\vec r|} ## ...(2) Then, magnitude of...

Hii, As we know, Scaler triple product is volume of parallelopiped constructed by its three sides. Similary, What is the physical significance and geometrical interpretation of Vector triple product ? Also, What are the application where we use such mathematics and why ? Regards, Rahul

I have been reading Ramamurti Shankar's book "Principles of Quantum Mechanics". The author, in the first chapter, briefs out the elementary mathematics required for quantum mechanics. Now, the author has described vector spaces, and made it very clear that only arrowed vectors that one studies...

Hi all, Suppose we have vectors coming in order as A, B and then C (but A must be deleted before C comes in). Then how to get the dot product between A and C? It is allowed to store some calculations of A before deleting elements of A, for example, we could store norm of A, dot(A, B) and etc...

Homework Statement 23. In a ABCD quadrilateral let P,Q,R,S be midpoints of sides AB,BC,CD and DA. Let X be the intersection of BR and DQ, and let Y be the intersection of BS and DP. If ##\vec{BX}=\vec{YD} ## show that ABCD is a parallelogram . Homework Equations ## (\vec{a}\cdot\vec{b})=0##...

Homework Statement If a, b and c are coplanar vectors related by λa+μb+νc=0, where the constants are non-zero, show that the condition for the points with position vectors αa, βb and γc to be collinear is: λ/α + μ/β + ν/γ = 0 Homework Equations Dot product Cross product Tripple product Vector...

This is the problem: Suppose a, b and c are linearly independent vectors. Determine whether or not the vectors (a + b), (a - b), and (a - 2b + c) are linearly independent. Here was my solution, which involved writing words (and hasn't actually been confirmed correct yet): Let's align a, b and...

If we use n linearly independent vectors x1,x2...xn to form a vector space V and use another set of n linearly independent vectors y1,y2...yn to form a vector space S, is it necessary that V and S are the same? Why? If we have a vector space Q that the dimension is n, can we say that any set of...

Homework Statement A line is given by the equation ##x + 2y - 3z = 7##. Find any vector in the direction parallel to this line in the Cartesian coordinate system. Homework Equations I imagine that there are some fundamental relationships I am missing here that would make this more...

Homework Statement Three ships A, B, and C move with velocities \vec{v_{1}} \ \vec{v_{2}} \ \vec{u} respectively. The velocities of A and B relative to C are equal in magnitude and perpendicular. Show that \left | \vec{u} -\frac{1}{2}(\vec{v_{1}} + \vec{v_{2}}) \right |^{2} = \left |...

Homework Statement given two unit vectors a= cosθi + sinθi b=cosΦi+sinΦj prove that sin(θ-Φ)=sinθcosΦ-cosΦsinθ using vector algebra[/B]Homework Equations sin(θ-Φ)=sinθcosΦ-cosΦsinθ[/B]The Attempt at a Solution axb= (cosθsinΦ-cosΦsinθ)k and I'm guessing that the change in sign has...

So, this seemed really fun to me until I got stuck. THE TASK is about an object with mass m, moving in a basic (2D) coordinate system. It is attached to origo (0, 0) by a "rope" with constant length r=5. In position P0(-5, 0) it has the velocity v0=[0, -10]. Hence, the object moves around origo...

Homework Statement The moment of the couple is 600k (N-m). What is the angle A? F = 100N located at (5,0)m and pointed in the positive x and positive y direction -F = 100N located at (0,4)m and pointed in the negative x and negative y direction Homework Equations M = rxF M = DThe Attempt at a...

Question: At what angles must be the two forces ##\vec A+\vec B## and ##\vec A-\vec B## act so that the resultant may be : $$\sqrt{ A^2+B^2}$$ Attempt at solution : Let the given forces be ##\vec F_1=\vec A+\vec B## and ##\vec F_2=\vec A-\vec B## . Now, Resultant vector : ##\vec F_1 + \vec...

I'm looking at: http://arxiv.org/pdf/gr-qc/9712019.pdf, deriving the FRW metric, and I don't fully understand how the Ricci Vectors eq 8.5 can be attained from 7.16, by setting ##\partial_{0} \beta ## and ##\alpha=0## I see that any christoffel symbol with a ##0## vanish and so so do any...

Homework Statement The velocity of an object as a function of time is given by: Vx(t) = 12t2 - 5t + 40 m/s Vy(t) = 5t - 30 m/s What is position at 2 sec if the object has an initial position of x = 5 m and y = 8 m? What is the instantaneous acceleration at 10 s? Homework EquationsThe...

How to find out the position vector of the centroid of tetrahedron , the position vectors of whose vertices are a,b,c,d respectively. I am familiar with the result, namely a+b+c+d/4 but want to know how to derive it without using the 3:1 ratio property. Any help would be appreciated. Thank you.

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three unit vectors such that $\left|\vec{a}+\vec{b}+\vec{c}\right|=\sqrt{3}$ and $\left(\vec{a}\times\vec{b}\right)\cdot \left(\vec{b}\times\vec{c}\right)+\left(\vec{b}\times\vec{c}\right)\cdot...

Problem: Let $\vec{a}$,$\vec{b}$ and $\vec{c}$ be non-coplanar unit vectors, equally incline to one another at an angle $\theta$. If $\vec{a}\times \vec{b} + \vec{b}\times \vec{c}=p\vec{a}+q\vec{b}+r\vec{c}$. Find scalars $p$,$q$ and $r$ in terms of $\theta$. Attempt: Taking the dot product on...

Problem: Consider the non zero vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ and $\vec{d}$ such that no three of which are coplanar then prove that $\vec{a}\left[\vec{b} \vec{c} \vec{d}\right]+\vec{c}\left[\vec{a} \vec{b} \vec{d}\right]=\vec{b}\left[\vec{a} \vec{c} \vec{d}\right]+\vec{d}\left[\vec{a}...

Problem: Given four non-zero vectors $\vec{a},\vec{b},\vec{c}$ and $\vec{d}$, the vectors $\vec{a},\vec{b}$ and $\vec{c}$ are coplanar but not collinear pair by pair and $\vec{d}$ is not coplanar with vectors $\vec{a},\vec{b}$ and $\vec{c}$ and...

Homework Statement this is an Apostol problem in chapter 12 and I guess it's a hypothetical definition of a norm of a vector Assuming this different definition of the norm prove these statements- Def. ||A||=\sum_{k=1}^{n}|a_{k}|, prove ||A||>0, if ||A||\neq0,||A||=0 if A=0...