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






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. brochesspro

    Find the unit vectors perpendicular to the given vectors

  2. 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## ...Then...
  3. 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...
  4. 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...
  5. 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?
  6. 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 ?
  7. 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...
  8. M

    Engineering Solving Problems Involving 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...
  9. 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...
  10. 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 =...
  11. 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?
  12. Blackbear38

    Using Inner Product Properties to Solve Vector Problems

    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) -...
  13. rudransh verma

    B Problem involving unit vectors

    For example is this correct : 19icap.4(-i cap) = 76(i.-i)= 76 Or is it , take - out. Then -76(icap.icap)= -76 Is it -76 or 76 ?
  14. A

    Subtracting two vectors -- I'm not getting the right answer

    I added x and y-axis so it would be square, and then vector bx would be same as vector a, but a didn't get it right. I am out of ideas. Can you help me?
  15. H

    Adding vectors in this 3-D problem

    I gathered that the final position of the vectors when added up would be (M,-M,M), but I'm not sure if this is correct.
  16. Istiak

    Find velocity with vector or without vector

    At the moment he wrote that ##\frac{1}{2}mv_2^2=\frac{1}{2}m(-\dot{y}+\dot{x})^2## But, I know from vector ##v_2=\sqrt{(-\dot{y})^2+(\dot{x})^2}##. At first I (he) found that ##v_2=-\dot{y}+\dot{x}##. But, when thinking of simple velocity in ##x## and ##y## coordinate then I get...
  17. Istiak

    Deriving the Differential Position Vector in Cylindrical Coordinates

    I had an equation. $$T=\frac{1}{2}m[\dot{x}^2+(r\dot{\theta})^2]$$ Then, they wrote that $$\mathrm dr=\hat r \mathrm dr + r \hat \theta \mathrm d \theta + \hat k \mathrm dz$$ I was thinking how they had derived it. The equation is looking like, they had differentiate "something". Is it just an...
  18. Monsterboy

    Vectors along different coordinate axes

    The answer in the textbook are options A, C and D. I understand why it is option A, because it is a scalar, I also get that option D is correct because the magnitude of a vector doesn't depend on the coordinate axes. I don't get how option C could be correct. If option C is correct why not D as...
  19. Istiak

    Understanding Direction of Unit Vectors r roof & phi roof

    The unit vector r roof points in the direction of increasing r with phi fixed; phi roof points in the direction of increasing phi with r fixed. Unlike x roof, the vectors r roof and phi roof change as the position vector r moves. What I was thinking of the image is Although, I was thinking why...
  20. A

    I Expressing Vectors of Dual Basis w/Metric Tensor

    I'm trying to understand why it is possible to express vectors ##\mathbf{e}^i## of the dual basis in terms of the vectors ##\mathbf{e}_j## of the original basis through the dual metric tensor ##g^{ij}##, and vice versa, in these ways: ##\mathbf{e}^i=g^{ij}\mathbf{e}_j##...
  21. B

    MHB What is the Angle Between Vectors Using the Dot Product Formula?

    Find the angle between the vectors v=-5\sqrt{3}i+5j and w=5i
  22. Rubberduck2005

    Tricky conceptual Projectile motion question

    So far all I have determined is the equations of motion for the two and that is as follows. It is trivial that y(t)=v1sin(Q)t -gt^2/2 and that x(t)=v2cos(Q)t. Now the angle that is anticlockwise from the negative horizontal of the robber is 90 - Q using basic trigonometry, using this we can...
  23. LCSphysicist

    How Do You Compute Killing Vectors for a Given Metric?

    I want to find all the killing vectors of the metric ##x²dx² + xdy²##. We could guess somethings by intuition and check it, but i decided to use the equation itself. Unfortunatelly, i realized that i am not sure how to manipulate the equation $$L_{\chi}g_{ab} = g_{ad}\partial_{b}...
  24. WMDhamnekar

    MHB Velocity and acceleration vectors and their magnitudes

    How to answer this question? I am working on this question. Any math help, hint or even correct answer will be accepted.
  25. snoopies622

    I How do I add "rotation" vectors (pseudo-vectors?)

    I'm tutoring an intro to meteorology pupil and learning about the conservation of potential vorticity, and realizing that I don't understand some basic rotational mechanics. For example, suppose I stand on the North Pole and hold a wheel such that the wheel's axis of rotation is parallel to the...
  26. Arman777

    A Calculating Lie Derivatives for Tensors & Vectors

    I am writing a code to calculate the Lie Derivatives, and so far, I have defined the Covariant derivative 1) for scalar function; $$\nabla_a\phi \equiv \partial_a\phi~~(1)$$ 2) for vectors; $$\nabla_bV^a = \partial_bV^a + \Gamma^a_{bc}V^c~~(2)$$ $$\nabla_cV_a = \partial_cV_a -...
  27. Arman777

    A Finding Killing Vectors of FLRW Metric: Simple Equation?

    I need to find the killing vectors of the FLRW metric. However, it seems that they are complicated. Is there a simple/general equation that gives the killing vectors for a given metric? Or do I have to solve ten independent killing equations simultaneously to find the killing vectors?
  28. K

    B Magnitudes of the sum of two vectors

    This is a question that I saw in a textbook: "If the magnitude of a+b equals the magnitude of a+c then this implies that the magnitudes of b and c are equal. Is this true or false?" The textbook says that this statement is true, but I'm inclined to believe it is false. I made a quick sketch to...
  29. mcastillo356

    Unit vectors -- How can they be dimensionless?

    Hi, what is a unit vector? I mean, it is ##\hat{A}=\vec A/|A|##. A dimensionless vector with modulus (absolute value) one, I've read somewhere. So, dimensionless with modulus. Isn't that a contradiction? I mean, absolute value regardless dimension? Am I out of context?. ##\Bbb R^3## is a...
  30. G

    Vectors - Aircraft's velocity relative to ground

    I attach my working below - my angle is correct according to mark scheme but magnitude isn't (should be 230). I think it's odd that my resultant velocity on a windy day is larger than velocity in still air, but apparently my angle is correct? I've been told that I've calculated the airspeed it...
  31. W

    I Trouble understanding contravariant transformations for vectors

    Hey, so I've been studying some math on my own and I'm really confused by this one bit. I understand what contravariant components of a vector are, but I don't understand the ways in which they transform under a change of coordinate system. For instance, let's say we have two coordinate...
  32. I

    Vectors - finding coordinates of collision point

    For car 1, the parametric equations are x = 1 + 0.8t and y=t. For car 2, the parametric equations are x=0.6s and y=2+s. (Let t and s represent time). Solving the system of equations, when the x values are equated are the y values are equated, I get s = -13 and t = -11. I assume that the 2 cars...
  33. Arman777

    Tensor Calculations given two vectors and a Minkowski metric

    Let us suppose we are given two vectors ##A## and ##B##, their components ##A^{\nu}## and ##B^{\mu}##. We are also given a minkowski metric ##\eta_{\alpha \beta} = \text{diag}(-1,1,1,1)## In this case what are the a) ##A^{\nu}B^{\mu}## b) ##A^{\nu}B_{\mu}## c) ##A^{\nu}B_{\nu}## For part (a)...
  34. karush

    MHB Q01 are linearly independent vectors, so are....

    Let A be invertible. Show that, if $\textbf{$v_i,v_2,v_j$}$ are linearly independent vectors, so are \textbf{$Av_1,Av_2,Av_3$} https://drive.google.com/file/d/1OuHxfUdACbpK4E5aca2oBzdaxGR0IYKv/view?usp=sharing ok I think this is the the definition we need for this practice exam question...
  35. karush

    MHB Creates vectors for the span

    ok I am trying to solve some other problems following this example but can[t see how the $z_1,z_2,z_3$ are created I know it is pulled for REFF matrix
  36. Pouramat

    Einstein's Vacuum Exploring the Metric & Killing Vectors

    Einstein's vacuum solution metric: $$ ds^2 = -(1-\frac{2GM}{r})dt^2 +(1-\frac{2GM}{r})^{-1}dr^2+r^2 d\Omega^2 $$ which ##g_{\mu \nu}## can be read off easily. metric Killing vectors are: $$ K = \partial_t $$$$ R = \partial_\phi $$ How can I relate these to Maxwell equation?
  37. S

    Physics History (Maxwell) Rotary Vectors?

    I am reading the text 'Innovations in Maxwell's Electromagnetic Theory'. on page 44 there is a discussion on Ampere's circuital law . The passage is below. I don't understand the final statement. "In general represent a kind of relationship that obtains between certain pairs of phenomena , of...
  38. E

    B What is the cross product of two vectors?

    I understand that dot product gives us a number and cross product gives a vector. Why is this vector orthogonal to the others two, and why it has magnitude |a|*|b|*sinΘ? How to use cross product? What does it give to us?
  39. karush

    MHB Linearly Dependent Vectors: Find h Value and Justify

    $\tiny{311.1.7.11}$ ok I am going to do several of these till I get it... Find the value(s) of h for which the vectors are linearly dependent. Justify $\left[\begin{array}{rrrrrr} 2\\-2\\4 \end{array}\right], \left[\begin{array}{rrrrrr} 4\\-6\\7 \end{array}\right], \left[\begin{array}{rrrrrr}...
  40. karush

    MHB 311.1.5.14 Use vectors to describe this set as a line in R^4

    ok, just now looking at some examples of how to do this $x_4$ is just a row with all zeros
  41. C

    Linear independence of Coordinate vectors as columns & rows

    Summary:: x Question: Book's Answer: My attempt: The coordinate vectors of the matrices w.r.t to the standard basis of ## M_2(\mathbb{R}) ## are: ## \lbrack A \rbrack = \begin{bmatrix}1\\2\\-3\\4\\0\\1 \end{bmatrix} , \lbrack B \rbrack = \begin{bmatrix}1\\3\\-4\\6\\5\\4 \end{bmatrix}...
  42. L

    Vectors and Trigonometry Homework Help

    Hey, I am new to this community and I am in need of help with this physics problem. I have used the formula above and the answer I get is 1.43s. The correct two answers are 0.68s and 2.4s. For the Vf the answer is 8.3 m/s.
  43. P

    Help graphing Vectors in polar form

    The equation I'm trying to graph on desmos is this with A & B as numbers, but I'm unsure how as it is a vector. r = (A cosθ sinθ cscθ - B sinθ cscθ) i + (A cosθ sinθ cscθ + B sinθ cscθ) j
  44. O

    How to find d2 when given d1 and d, total time, and average velocity?

    I rearranged the displacement formula to d2 = d + d1. I used cosine law to solve for d2 since the triangle is not right-angled but I am not getting the correct answer or angle for d2. The angle I used in cosine law (based on the diagram) was 32+12+90 = 134. d = v(t) = 130(3) = 390 km/h [N 32 E]...
  45. LCSphysicist

    Condition to three vectors being collinear

    Now i am rather confused, the answer apparently is that ##(w-u) = \lambda(u-v)## But, i could find a way that disprove the answer, that is: Be u v and w vectors belong to R2, a subspace of R3: What do you think? This is rather strange.
  46. S

    Converting State Vectors to Keplerian Orbital Elements for Binary Objects

    Homework Statement:: I'm working on a personal project to convert objects from a simulation using state vectors for position and velocity to Keplerian orbital elements (semimajor axis, eccentricity, argument of periapsis, etc.). However, the equations I am using do not calculate the...
  47. Kaguro

    I Exploring the Connection Between Dual Vectors and Covectors in Vector Spaces

    I understand that a vector space is a set of objects closed under addition and scalar multiplication and satisfies several properties. A functional is a map that takes a vector and produces a scalar. A functional is also called a dual vector. A covector is an object which transforms via the...
  48. greg_rack

    Calculating vector cross product through unit vectors

    Writing both ##\vec{U}## and ##\vec{B}## with magnitude in all the three spatial coordinates: $$ \vec{U}\times \vec{B}= (U_{x}\cdot \widehat{i}+U_{y}\cdot \widehat{j}+U_{z}\cdot \widehat{k})\times (B_{x}\cdot \widehat{i}+B_{y}\cdot \widehat{j}+B_{z}\cdot \widehat{k})$$ From this point on, I...
  49. T

    Forces on a body with vectors (IWTSE.org)

    I think I've made a mistake with the vectors. I was trying to get to the same point as the original working (made by IWTSE, not me), but one of the sides of a triangle can't be 0. I would appreciate some help.
  50. binbagsss

    MATLAB Simple addition of vectors of different lengths

    I want it to be able to add vectors of different length with a 'shift' of c, and where elements do not align to assume a zero in the place of that, My function is: %c is the parameter shift function[sum]=vecadd(y,b,c) p=length(input) q=length(a) if p > q for i= 1+delay:p+delay...
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