# Vector algebra Definition and 34 Discussions

In mathematics, vector algebra may mean:

Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space.
The algebraic operations in vector calculus, namely the specific additional structure of vectors in 3-dimensional Euclidean space

R

3

{\displaystyle \mathbf {R} ^{3}}
of dot product and especially cross product. In this sense, vector algebra is contrasted with geometric algebra, which provides an alternative generalization to higher dimensions.
An algebra over a field, a vector space equipped with a bilinear product
Original vector algebras of the nineteenth century like quaternions, tessarines, or coquaternions, each of which has its own product. The vector algebras biquaternions and hyperbolic quaternions enabled the revolution in physics called special relativity by providing mathematical models.

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1. ### A Converting this vector into polar form

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)...
2. ### Vector Kinematics -- A particle leaves the origin with an intial velocity and experiences constant acceleration

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.
3. ### Roundtrip by Plane

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...

5. ### I Equation for circle points in 3D

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...
6. ### Show the identity ##\vec{\nabla}(\vec{r} \cdot \vec{u})##

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}...
7. ### I Axes of the 2-d coordinate system used in vector resolution

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...
8. ### Curve integrals, del operator

My attempt is below. Could somebody please check if everything is correct? Thanks in advance!
9. ### Curvilinear coordinate system: Determine the standardized base vectors

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!
10. ### To find the time to nearest appoach of a dinghy and a buoy

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...
11. ### To find the closest approach of two ships

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...
12. ### I Velocity Vector Transformation from Cartesian to Spherical Coordinates

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...
13. ### Determining whether a set is a vector space

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...

21. ### Find the angle between 2 vectors

Homework Statement |a| = 2 |b| = ## \sqrt3## |a - 2b| = 2 Angle between a and b Homework Equations The 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 +...
22. ### Difficult version of boatman problem

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...
23. ### I Can you reduce a vector triple product? i.e. (A x (uB x C))

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...
24. ### Calculating angle between velocity and position vector

Homework Statement Homework Equations The 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...
25. R

### I Vector Triple Product - Physcial Significance

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
26. ### B What is the requirement for something to qualify as vector?

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...
27. ### Dot Product of Streaming vectors

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...
28. ### Vector algebra (proving you have a parallelogram by using vectors)

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##...
29. ### Colinear points in a vectorplane

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...
30. ### How to correctly solve this problem? (linear dependency)

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...
31. ### I Same vector space for arbitrary independent vectors?

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...
32. ### Advanced Vector Problem: Ships

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 |...
33. ### Task: Function for the acceleration throughout a loop?

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...
34. ### Angle between coupled forces

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 = D The Attempt...