Vectors Definition and 1000 Threads

In mathematics, physics and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a ray (a directed line segment), or graphically as an arrow connecting an initial point A with a terminal point B, and denoted by






A
B

→




{\displaystyle {\overrightarrow {AB}}}
.A vector is what is needed to "carry" the point A to the point B; the Latin word vector means "carrier". It was first used by 18th century astronomers investigating planetary revolution around the Sun. The magnitude of the vector is the distance between the two points, and the direction refers to the direction of displacement from A to B. Many algebraic operations on real numbers such as addition, subtraction, multiplication, and negation have close analogues for vectors, operations which obey the familiar algebraic laws of commutativity, associativity, and distributivity. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space.
Vectors play an important role in physics: the velocity and acceleration of a moving object and the forces acting on it can all be described with vectors. Many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances (except, for example, position or displacement), their magnitude and direction can still be represented by the length and direction of an arrow. The mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the coordinate system include pseudovectors and tensors.

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  1. brotherbobby

    B To calculate the polar unit vectors using rotated coordinates

    We know that if cartesian coordinates ##(x,y)## (see figure alongside) are rotated to ##(x',y')## about the origin by an angle ##\theta## counter-clockwise as shown, the rotated coordinates are given by $$\begin{align*}x'&=\cos\theta \;x+\sin\theta \;y\\ y'&=-\sin\theta \; x+\cos\theta \; y...
  2. A

    I Determining Whether Solution is Vector vs Scalar?

    Ok, so currently, I'm working on problems involving work done by a general variable force. I had a question as far as solving these that ties into other problems as well... I see in some of the worked problems in this textbook, when they take the integral of a force equation to determine the...
  3. snoopies622

    I What are Killing vectors?

    I don't have an intuitive feel for Killing vectors. Wikipedia says, " . . . more simply, the flow generates a symmetry, in the sense that moving each point of an object the same distance in the direction of the Killing vector will not distort distances on the object." That just sounds like...
  4. E

    What is the difference between the two sine rules for resultant vector?

    Hello can anyone help me with this: there are two sine rules for finding the direction of a resultant vector;one for the sides and one for the angle; I tested both formulas and they all worked well and gave me equal answers, does that mean I can use them interchangeably,the rules are: a/sinA =...
  5. Manish_529

    Change in the unit vectors

    i tried integrating the stuff but it didn't work what to do
  6. chwala

    Obtain an equation of the plane in the form ##px+qy+rz=d##

    The solution is here; Now to my comments, From literature, the cross product of two vectors results into a vector in the same dimension. A pointer to me as i did not know the first step. With that in mind and using cross product, i have ##(1-1)i - (-1-1)j+(1+1)k =0i+2j +2k## as shown in ms...
  7. F

    I Standard basis and other bases...

    Hello, I am review some key linear algebra concepts. Let's keep the discussing to 2D. Vectors in the 2D space can be simplistically visualized as arrows with a certain length and direction. Let's draw a single red arrow on the page representing vector ##X##, an entity that is independent of the...
  8. N

    B Confused about dot product multiplication

    I'm confused about what we are really measuring when taking the dot product of two vectors. When we say we are measure "how much one vector points in the direction of the other", that description is not clear. At first I thought it meant how much of a shadow one vector casts on another and I...
  9. A

    Given value of vectors a,b, b.c and a+(b×c), Find (c.a)

    I thought this was too easy $$a+(b\times c)=0\implies a=-(b\times c)=(c\times b)$$ Then $$3(c.a)=3(c.(c\times b))=0$$ Since cross product of vectors is perpendicular to both vectors and dot product of perpendicular vectors is zero. Now here's the problem, correct answer given is 10. But how do...
  10. nomadreid

    I Basic question about Bra-Ket notation

    This is so basic as to be embarrassing, but I haven't figured out my misunderstanding of some basic notation. [1] If v and w are two vectors in a Hilbert space, then <v|w> is interpreted as the probability amplitude of w collapsing into v. [2] However, if P is a projection, then <v|Pv>...
  11. MP97

    Complete the parametric equation for the line where the planes cross

    First, I use the unit vector of each plane, and I compute their cross-product to obtain a vector parallel to the line of interception. Then, I algebraically use x=0 to obtain the coordinates of the point in the line of interception. However, not having a y coordinate in plane one is confusing me.
  12. C

    I How to establish which side of a square a ray will intersect?

    Consider some ray ## \bar{r} ## that starts at point ## A=(a_x,a_y) ## and faces some direction and consider an upright square ( i.e. it's not rotated ) at some location: Question: if we let the ray continue until hitting the square, how can we detect which face of the square was hit? is there...
  13. brotherbobby

    To find the component of a vector perpendicular to another

    Problem statement : I copy and paste the (slightly different) problem statement as it appeared in the text to the right. Attempt : By inspection, we find that the vector ##\vec B'## perpendicular to ##\vec B = 3\hat i+4\hat j## is ##\boldsymbol{\vec B' = 4\hat i -3\hat j}##, remembering that...
  14. Rhdjfgjgj

    Writing correct sign of quantities

  15. kai sinclair

    Efficient Calculation of Final Magnitude and Angle in Vector Addition Method

    okay so I'm a Electrician I've found short method of calculating the final magnitude of a system (Lₜ), this relies mainly on Eucliud's axiom of angles within parrallel lines and is this ∑(cos(θₙ-θₜ)⋅Lₙ)=Lₜ where Lₙ and θₙ are the initail manignitude and angles respectively, and θₜ is the final...
  16. billard

    I Deriving formula for kinetic energy

    Hello! I am new to the differential version of classical physics, and I am trying to work how to derive kinetic energy from some pre-assumed equations: Assume that we know: ##\ddot{z} = 0## and ##m\ddot{\textbf{r}} \cdot \dot{\textbf{r}} = 0##This results in $$\frac{1}{2}m\dot{r}^2 = W =...
  17. Antarres

    I A question about geodesics and notion of parallelism

    I think I have a slight misconception maybe, but I was wondering about this question. Usually when we say that the vectors are parallel, we say that it means that there's an equation ##k = \alpha l##, for the vectors ##k## and ##l## and some scalar ##\alpha##. In the context of differential...
  18. J

    Standard angles for vectors (Mechanics)

    n/a
  19. P

    B Looking for help understanding scalars vs. vectors

    Why are pressure and energy not considered as vectors?
  20. chwala

    Determine the position vector of ##C##

    Highlighted part only... Part (a) was easy ##2\sqrt 5##. For part (b), ...##BC=4i+2j## it follows that, ##OC=OB+BC## ##OC=3i+5j+4i+2j=7i+7j## correct? any other better approach guys! For part (c), I will form the equations as follows; Let ##D(x,y)## then, ##x-4=2(4-3)## and...
  21. CaliforniaRoll88

    Does the book have an error in the addition of these two vectors?

    (a) $$\vec A = -\vec M+2\vec N=26\hat a_x+10\hat a_y+4\hat a_z$$ Unit Vector Formula $$\hat a_A=\frac{26\hat a_x+10\hat a_y+4\hat a_z}{\sqrt {26^2+10^2+4^2}}$$ $$0.923\hat a_x+0.355\hat a_y+0.142\hat a_z$$ The book gives ##0.92\hat a_x+0.36\hat a_y+0.4\hat a_z## Not sure how the book get 0.4 for...
  22. M

    Finding vectors parallel to a given vector

    For (b) of this problem, The solution is, However, I am confused why the two parallel vectors are ##(\frac{2}{\sqrt{13}}, \frac{3}{\sqrt{13}})## and ## (-\frac{2}{\sqrt{13}}, -\frac{3}{\sqrt{13}}) ## should it not be ##(2,3)## and ##(-2,-3)##. Do somebody please know why they wrote that? Also...
  23. TGV320

    I Law of Cosines in Linear Algebra: Understanding the Dot Product of Unit Vectors

    HI, I am studying linear algebra, and I just can't understand why "Unit vectors u and U at angle θ have u multiplied by U=cosθ Why is it like that? Thanks
  24. D

    A Understanding killing vectors and transformations of metric

    Hi, I am reading through my lecture notes - I haven't formally covered killing vectors but it was introduced briefly in lectures. Reading through the notes has highlighted something I am not sure about when it comes to co-ordinate transformations. Q1.Can someone explain how to go from...
  25. themethetion

    B Calculating the perfect tennis shot using vectors

    Context: I must develop a vector that models the path of a tennis ball using vectors without physics formulas I have developed a function that represents the perfect shot in tennis in terms of x & y where x is the court's length and y is the height. And then x & z where x is the court's length...
  26. themethetion

    Projectile Motion Using Vectors

    TL;DR Summary: Using vector functions how can I find the minimum average velocity (something greater than 11.86m/s) of a ball if the launch angle is unknown and if I have a point that the object must travel through (11.86, 3.47)? In my assignment, I developed a function for a lob shot...
  27. chwala

    Finding the angle between two vectors

    This is clear to me; i just wanted to know in which contexts is one allowed to use one rule over the other; or it does not matter. The angle i realise can also be found by; ##\sin θ = \dfrac{||v×w||}{||v||||w||}##=...
  28. M

    The direction of flux vectors in derivation of conservation of mass

    In the derivation of the conservation law of the conservation of mass, the flux on one side enters and the flux on the other side leaves the control volume. I presume this is due to the assumption that the volume is infinitesimally small and hence v(x,y,z,t) will not change directions...
  29. P

    B Question about orthogonal vectors and the cosine

    Hi, The orthogonality defect is ##\prod_i ||b_i|| / det(B)##. Now it is said: The relation between this quantity and almost orthogonal bases is easily explained. Let ##\theta_i## be the angle between ##b_i## and ##span(b_1,...,b_{i-1})##. Then ##||b_i^*|| = ||b_i|| cos(\theta_i)##. [...] So...
  30. nataelp

    Force between 2 point charges in VECTOR format (x i + y j + z k)

    I tried using the distance between r2 and r1 and plugging them into the equation for i, j, k. >> So for the force in the x direction it was k*(4E-6*4E-6)/(4-9)^2. The answer I got was wrong according to webassign. Can someone please tell me what I am missing?
  31. M

    Explaining the Cross Product for Two Vectors

    Hi! For this problem, The solution is, However, I don't understand their solution at all. Can somebody please explain their reasoning in more detail. Many thanks!
  32. A

    Two vectors forming right handed set?

    I am confused by a question. I thought "right handed set" only applied to sets of three vectors. However I have been given 2 vectors and asked "check whether they are perpendicular to each other and if they form a right handed set. If they don't form a right handed set, the second vector must be...
  33. homeworkhelpls

    I What does vector subtraction for two non-perpendicular vectors look like?

    vector subtraction of ppl is simple but i cant visualise the subtraction please help
  34. K

    I Understand 4-Vectors & Spacetime: Hartle Gravity Chapter 5

    Hartle, gravity. Chapter 5 "A four-vector is defined as a directed line segment in four-dimensional flat spacetime in the same way as a three-dimensional vector (to be called a three-vector in this chapter) can be defined as a direcied line segment in three-dimensional Euclidean Space"For...
  35. Math Amateur

    I Dirac Notation for Vectors and Tensors (Neuenschwander's text ....)

    I am reading Tensor Calculus for Physics by Dwight E. Neuenschwander and am having difficulties in confidently interpreting his use of Dirac Notation in Section 1.9 ... in Section 1.9 we read the following: I need some help to confidently interpret and proceed with Neuenschwander's notation...
  36. M

    I Are There Other Ways To Represent Vectors Besides Arrows?

    Hello. I wonder about the representation of vectors, so I wanted to ask: how many different ways vectors be represented? As far as I know two: Geometrically and Algebraically. There is only one way to represent vectors in geometrically: arrows, however there are several or more methods to...
  37. J

    Is it possible to have a vector space with restricted scalars?

    I don't understand the solution: that for (1, ..., 1) the additive inverse is (-1, ..., -1), so the condition is not satisfied (and it is not a subspace). Which condition is not met? Thank you.
  38. D

    Why does the dot product in this solution equal zero?

    Hi everyone I have the solutions for the problem. It makes sense except for one particular step. Why does the dot product of a and b equal zero? I thought this would only be the case if a and b were at right angles to each other. The solutions seem to be a general proof and should work for...
  39. tbn032

    B Confusion about the angle between two vectors in a cross product

    The magnitude of cross product is defined of vector A⃗ and B⃗ as |A⃗×B⃗|=|A⃗||B⃗|sinθ where θ is defined as the angle between the two vector and 0≤θ≤π.the domain of θ is defined 0≤θ≤π so that the value of sinθ remains positive and thus the value of the magnitude |A⃗||B⃗|sinθ also remain positive...
  40. H

    Vectors as geometric objects and vectors as any mathematical objects

    In geometry, a vector ##\vec{X}## in n-dimensions is something like this $$ \vec{X} = \left( x_1, x_2, \cdots, x_n\right)$$ And it follows its own laws of arithmetic. In Linear Analysis, a polynomial ##p(x) = \sum_{I=1}^{n}a_n x^n ##, is a vector, along with all other mathematical objects of...
  41. LCSphysicist

    Preservation of the angle between two vectors

    Homework Statement:: . Relevant Equations:: . Generally, when we talk about preservation of angle between two vectors, we talk about conformal transformation. But what is confusing me is, shouldn't any general transformation of coordinates preserve the angle between two vectors? What i mean...
  42. Salmone

    I Linear independence of three vectors

    If I've got three vectors ##\vec{a}##, ##\vec{b}## and ##\vec{c}## and ##\vec{a}##, ##\vec{b}## are linearly independent and ##\vec{c}## is linearly independent from ##\vec{a}##, is ##\vec{c}## also linearly independent from ##\vec{b}##?
  43. lola1227

    Momentum Collision Homework Problem -- help please

    Parallel: M1V1+M2v2=M1V1’+M2V2’ (0.5)(3)+0=(0.5)(cos60)(3)+V2’Cos(x)(0.5) V2’cos(x)= Perpendicular: M1V1+M2v2=M1V1’+M2V2’ 0=(0.5)(0.3)(sin60)+V2’sin(x)(0.5) V2’sin(x)= And the divide 2 by 1 Which is tan(x)=2/1 And then plug then back into solve, but I don’t think we do it like this because...
  44. X

    I Is tensor product the same as dyadic product of two vectors?

    Is tensor product the same as dyadic product of two vectors? And dyadic multiplication is just matrix multiplication? You have a column vector on the left and a row vector on the right and you just multiply them and that's it? We just create a matrix out of two vectors so we encode two...
  45. R

    Understanding Velocity Addition Laws for People on Train & Ground

    I kind of just made up the questions. I realize this is a basic question but my knowledge of physics is very limited.q1 answer v_left_ball = v_left_ball - v_train v_right_ball = v_right_ball + v_train q2 answer To get the speed from Bob's frame I would use v_Bob = v_Bob + v_Alice To get the...
  46. H

    I Sum of the dot product of complex vectors

    Summary:: summation of the components of a complex vector Hi, In my textbook I have ##\widetilde{\vec{E_t}} = (\widetilde{\vec{E_i}} \cdot \hat{e_p}) \hat{e_p}## ##\widetilde{\vec{E_t}} = \sum_j( (\widetilde{\vec{E_{ij}}} \cdot {e_{p_j}}*) \hat{e_p}## For ##\hat{e_p} = \hat{x}##...
  47. R

    I Why is momentum considered a vector and kinetic energy a scalar?

    I'm not interested in the mathematical derivation, the mathematical derivation already is based on the assumption that momentum is a vector and kinetic energy is a scalar, thus it proves nothing. Specifically, what happens if we discuss scalarized momentum? What happens if we discuss vectorized...
  48. chwala

    Find the shortest distance between the given vectors in 3D

    This is a textbook problem...the only solution given is ##3.##...with no working shown or given. My working is below; i just researched for a method on google, i need to read more in this area...use of the directional vector may seem to be a more solid approach. Ok i let ##A=(6,-4,4)##...
  49. Stewkatt

    2D kinematics -- Calculate the acceleration of the jumping athlete

    this is my work but the answers say 11 m/s^2 so I made an error somewhere. Also if someone could help me with solving the direction for the acceleration, that would be greatly appreciated.
  50. G

    A Understand (k,l) Tensors in Gen. Relativity

    In both Wald and Carroll, a type (k,l) tensor has k dual vectors and l vectors, yet a (1,0) tensor is a vector and a (0,1) tensor is a dual vector. I must be missing something simple. Please explain.
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