# What is Vector algebra: Definition and 73 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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18. ### B How to specify the direction of an area vector?

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...
19. ### Change of basis computation gone wrong....

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...
20. ### If given A,B,C, Find constants x,y |xA +yB=C

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...
21. ### Parallelism of Time-varying Vectors

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)...
22. ### Problem using vectors and relative motion

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...

41. ### Deriving FRW Metric: Ricci Vector Algebra Explained

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...
42. ### Vector algebra - The velocity is given by

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...
43. ### MHB Vector algebra- centroid of tetrahedron

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.

47. ### MHB Vector Algebra: Finding the Angle Between Two Vectors

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...
48. ### Proving Vector Norm Properties Using Different Definitions

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...
49. ### Help with vector algebra getting an undefined answer

Homework Statement Determine the magnitude of F that will cause the net effect of the three forces to act vertically. In the diagram there are three bolts (I'll refer to the first two as B1 and B2), and the third bolt, F, is undefined. F acts 45 degrees northeast of the horizontal, B1 is 410N...
50. ### Wave Vector Algebra: Proving k = 2∏/λ

Homework Statement I'm having difficulty showing that the general equation k = 2∏/λ holds from the component equations kx = 2∏/λx etc..., k = √(kx2 + ky2 + kz2), and λ = √(λx2 + λy2 + λz2). Any help? There is a photo from a textbook with the equations also below. Homework Equations...