What is Vectors: Definition and 1000 Discussions

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


{\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. 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 =...
  2. 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...
  3. J

    Standard angles for vectors (Mechanics)

  4. P

    B Looking for help understanding scalars vs. vectors

    Why are pressure and energy not considered as vectors?
  5. 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...
  6. 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...
  7. C

    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...
  8. TGV320

    I 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
  9. 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...
  10. 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...
  11. 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...
  12. 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||}##=...
  13. 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...
  14. 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...
  15. 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?
  16. C

    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!
  17. 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...
  18. 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
  19. 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...
  20. 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...
  21. M

    I Representation of Vectors?

    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...
  22. J

    Sets of Vectors as Subspaces

    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.
  23. 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...
  24. 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...
  25. 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...
  26. 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...
  27. 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}##?
  28. 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...
  29. 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...
  30. 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...
  31. 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}##...
  32. 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...
  33. 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)##...
  34. 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.
  35. 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.
  36. runningphysics

    Two forces acting on an object given in vectors - SOLVED

    I tried splitting the forces up into F1 and F2 making Newtons second law equation into F1+F2=ma. Then I added over the the first force given. multiply the mass to the acceleration terms to get F2= (m*ai + m*aj) - F1
  37. JackFyre

    B A doubt regarding vectors, scalars and their role in physics

    I have a doubt regarding the basic function of vectors and scalars in physics- What is the guarantee that every quantity(measured) in physics can be classified as either a vector or a scalar, and that while performing operations on said quantities, they will obey the already established rules...
  38. guyvsdcsniper

    Prove vectors are orthogonal

    I feel like this question is very straight forward and my explanation below summarizes the answer pretty well. Could someone confirm this or tell me if I am missing something? We have V which is a vector, but the question states it is a constant. If I take the derivative of V, represented by...
  39. brochesspro

    Find the unit vectors perpendicular to the given vectors

  40. chwala

    Solve these simultaneous equations that involve vectors

    Find the question and solution here; Ok, i was able to solve this by using, ##3A=3ax+12ay+6bx+3by+3b## ##2B=2ay-4ax+4a+4bx-6by-2b## leading us to the simultaneous equation; ##7x+10y=4## ##2x+9b=-5## ##x=2## and ##y=-1## I had initially tried the approach of using ##3A=2B## →##B=1.5A##...
  41. e2m2a

    I Confusion between vector components, basis vectors, and scalars

    There is an ambiguity for me about vector components and basis vectors. I think this is how to interpret it and clear it all up but I could be wrong. I understand a vector component is not a vector itself but a scalar. Yet, we break a vector into its "components" and then add them vectorially...
  42. H

    I S is set of all vectors of form (x,y,z) such that x=y or x =z. Basis?

    ##S## is a set of all vectors of form ##(x,y,z)## such that ##x=y## or ##x=z##. Can ##S## have a basis? S contains either ##(x,x,z)## type of elements or ##(x,y,x)## type of elements. Case 1: ## (x,x,z)= x(1,1,0)+z(0,0,1)## Hencr, the basis for case 1 is ##A = \{(1,1,0), (0,0,1)##\} And...
  43. rudransh verma

    What kind of motion is this? (bicycle velocity vectors)

    How the heck the bicycle reach from point A to point B with that velocity vector directions?
  44. Danielle46

    I have to prove that vectors in spherical coordinates are clockwise

    I should use the cross product but I don´t know how. I tried to calculate it but it didn´t work out as expected. Please can you give me one example how to do it ?
  45. mohammed

    Solving Electric Field Vector Problems Using Gauss's Law

    I'm preparing for exam but it seems I can't find problems similar to this on the internet. Here I will apply Gauss's law on the electric field vector to get the charge density. but the problem is that I can't find similar examples on the internet that uses direct vectors on Maxwell's equations...
  46. M

    Engineering Manipulating complex vectors

    Hi Here is my attempt at a solution for problems 1) and 2) that can be found within the summary. Problem 1) a = 3-2i b= -6-4i c= 4+ 6i d= -4+3i Now, to calculate each vector modulus, I applied the following formula: $$\left| Vector modulus \right| = \sqrt {(a^2 + b^2) }$$ where a = real part...
  47. U

    Vectors in yz and xz plane dot product, cross product, and angle

    I tried to find the components of the vectors. ##a_y =2.60 sin 63.0 = 2.32## and assuming the z axis would behave the same as an x-axis ##a_z =2.60 cos 63.0 = 1.18## ##b_z =1.30 sin 51.0 = 1.01## making the same assumption ##b_x =1.3 cos 51.0 = 0.82## I now think I should have switched these...
  48. U

    Displacement problem with unit vectors

    (a) I did (7.07*4.1)-(-7.03*3.94)=56.7 with this method I got this answer correct in my first attempt. (b) This where I seem to have gone wrong. I used a · b = (axbx +ayby) then I used a = sqrt(ax2+ay2) to get a single number for the answer. Filling in the numbers 7.07*-7.03 + 4.94*4.1 =...
  49. E

    I finding resultants using sine/cosine law

    I need help finding the resultant with vectors: 37.5N[NE] and 45N[21° S of E] I just don't know a way to find the angles within this triangle to help me get the resultant, so can anybody help me out?
  50. Blackbear38

    Dot product vectors problem

    Summary:: I need to solve a problem for an assignment but just couldn't find the right approach. I fail to eliminate b or c to get only the magnitude of a. Let a, b and c be unit vectors such that a⋅b=1/4, b⋅c=1/7 and a⋅c=1/8. Evaluate (write in the exact form): - ||4a|| - 3a.5b - a.(b-c) -...