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. Crush1986

    Showing Complex Vectors are Orthonormal

    Homework Statement let \epsilon_1 and \epsilon_2 be unit vectors in R3. Define two complex unit vectors as follows: \epsilon_{\pm} = \frac{1}{\sqrt{2}}(\epsilon_1 \pm i \epsilon_2) verify that epsilon plus minus constitutes a set of complex orthonormal unit vectors. That is, show that...
  2. L

    3-D vectors forces exerted upon a mast

    Homework Statement The thick arrows represent forces exerrted upon the mast Let ## \vec{r_2} = ~~the~~ longer~force~ vector~~## ## \vec{r_1} = ~the ~shorter ~force ~ vector## correspondinly ## \vec{R_2} = cable ~~2## ## \vec{R_1} = cable~~1 ## r_1 is in the same direction as R_1 r_2 is in...
  3. Z

    I Do normal differentiation rules apply to vectors?

    (I am not very sure if this is a high-school level question or a undergraduate level question. Sorry.) Does our normal differentiation rules, like the product rule and quotient rule apply to vectors? Say for example, differentiate ##r \times \dot r## ##r## is radius vector, ##\dot r## is the...
  4. Vitani11

    Bra ket notation for magnitude of two vectors

    Homework Statement If I had two vectors say ⟨em|f⟩⟨f|em⟩ does this equal |⟨em|f⟩|2? e is a basis and f is some arbitrary function. I ask this because I have a problem which is to show the following: Show that for the Fourier expansion of |f⟩ in terms of Fourier basis vectors |em⟩ is...
  5. donaldparida

    B Understanding the Direction of Axial Vectors in Rotational Motion

    Background: I have just started studying vector algebra. I studied that physical quantities like velocity, force, momentum, etc. are vectors (My book addresses them as polar vectors/true vectors) and physical quantities like quantities like angular velocity, torque, angular momentum are also...
  6. arpon

    I Is Second rank tensor always tensor product of two vectors?

    Suppose a second rank tensor ##T_{ij}## is given. Can we always express it as the tensor product of two vectors, i.e., ##T_{ij}=A_{i}B_{j}## ? If so, then I have a few more questions: 1. Are those two vectors ##A_i## and ##B_j## unique? 2. How to find out ##A_i## and ##B_j## 3. As ##A_i## and...
  7. itamar123

    Why are the traction vectors on each surface independent?

    Hey y'all, my first thread here, Got a burning question that has been disturbing my serenity. In all derivations of the stress tensor that I've seen they didn't explain it that much, So my question is, why do the traction vectors on each surface are independent? From what I understood, the...
  8. andylatham82

    I How to find angle between vectors from dot and cross product

    Hi, hopefully a quick question here...how do you calculate the angle between two vectors if the only information you have is the value of their scalar product and the magnitude of their cross product? Thanks! Andy
  9. jlmccart03

    Identifying Magnetic Interactions

    Homework Statement There are two problems: 1. At the instant the switch is closed determine the direction of the net force exerted by the magnet on the wire segment at the instant that the magnet is in the position shown. Explain. 2b. Suppose the a third wire, carrying another current i0 out...
  10. andylatham82

    I Angle between vectors via scalar product vs vector product

    Hello, I have a question about why I can't determine the angle between two vectors using their cross product. Say there are two vectors in the XY-plane that we want to find the angle between: A = -2.00i + 6.00j B = 2.00i - 3.00j The method to do this would be to work out the scalar product of...
  11. R

    I Integrating a curve of position vectors

    I'm looking at different ways to express the derivative a curve, like circular and tangent/normal components. Is there no such way that let's you express a vector integral in terms of information from the vector you want to integrate?
  12. L

    MATLAB How to Multiply Two Vectors in MATLAB with Specific Conditions?

    Hi--I have two vectors ##x=(x_1, x_2, ..., x_n)## and ##y=(y_1, y_2, ..., y_n)##. Now I want them to be multiplied in the following way: for each ##i=1,2,..,n##, I need ##x_i*y_{i-1}-y_n##. Can anyone help me on how to code this in Matlab? BTW, I also want to input the length of the two...
  13. G

    I Determining Vector Direction: Finding Unit Vectors

    Why is there a need to find unit vector? If we are given a vector we can always find its direction.
  14. Y

    C/C++ [C++] How to return and call vectors from functions?

    Hi, beginner coder here. I have a somewhat solid understanding of both vectors and functions, and have used the two of them many times, but I'm have trouble coding functions that have vectors in their parameters and as their return values. Another thing I'm having trouble with is calling the...
  15. Ahmed Elhossiny

    I Why Use Position Vectors: Exploring Benefits of Different Coordinates

    Hello there, get the notion of position vectors for a particle, but why we use it instead of cartisean coordinates XYZ? What info does the vectors tell us that the cartisean coordinates doesn't tell us? So if we say a point has coordinates x=2, y=3, z=5 We say its position vector is r=i 2 +...
  16. O

    A Pullback and Pushforward in Manifolds: Why Do We Do It?

    In my ignorance, when first learning, I just assumed that one pushed a vector forward to where a form lived and then they ate each other. And I assumed one pulled a form back to where a vector lived (for the same reason). But I see now this is idiotic: for one does the pullback and pushforward...
  17. Minal

    A Find vectors in Orthogonal basis set spanning R4

    An orthogonal basis set spanning R4 has four vectors, v1, v2, v3 and v4. If v1 and v2 are [ −1 2 3 0 ] and [−1 1 −1 0 ] find v3 and v4. Please explain this in a very simple way.
  18. Grey_Thunderhead

    Find ground speed and direction of airplane

    Homework Statement [/B] "An airplane is flying on a bearing of 340 degrees at 400mph. A wind is blowing at a bearing of 320 degrees at 30mph. Find ground speed and direction of the plane." Homework Equations vx=vcosϕ vy=vsinϕ The Attempt at a Solution [/B] First, my teacher told us to change...
  19. davidge

    I Exploring 2D Translational and Rotational Killing Vectors in Polar Coordinates

    Hi everyone. What are the components of the 2 Translational Killing Vectors in 2-dimensions, in Polar Coordinates? I've solved the Killing equation using Maple, and the solution was ##\xi_r = 0##, ##\xi_{\theta} = r^2##, but I guess that these are the components for the rotation Killing Vector...
  20. Jovy

    What is the sine of the angle between two given vectors?

    Hello, I apologize in advance for the way this post looks. I am new to this forum and I've never used LaTeX Primer. I noticed that someone has prevoiusly asked the same question, but I still do not understand how to get to the answer. Also, I tried posting an image but I could not; and this...
  21. M

    Mechanics question: Roller passing over a step

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  22. S

    Vectors and Kayaking: Solving for Velocity, Time, and Distance

    Hello, Could someone please check my answer to this problem involving vectors. I have a feeling my answers are totally off; especially for B and C. All my work and applicable equations are shown in the pictures below. If my answer is wrong, please provide me with guidelines for solving the...
  23. F

    Conversion vectors in cylindrical to cartesian coordinates

    Homework Statement It's just an example in the textbook. A vector in cylindrical coordinates. A=arAr+aΦAΦ+azAz to be expressed in cartesian coordinates. Start with the Ax component: Ax=A⋅ax=Arar⋅ax+AΦaΦ⋅ax ar⋅ax=cosΦ aΦ⋅ax=-sinΦ Ax=ArcosΦ - AΦsinΦ Looking at a figure of the unit vectors I...
  24. binbagsss

    A Flat s-t 4d killing vectors via solving killing equation

    So I know what these are 4 translation : ##\frac{\partial}{\partial_ x^{u}} = \partial_{x^u}## 3 boost: ##z\partial_y - y \partial_z## and similar for ##x,z## and ## y,x## 3 rotation: ##t\partial_x + x\partial_t ## and similar for ##y , z## however I want to do it by solving Killing equation...
  25. The black vegetable

    Cartesian Coordinates and Cross Product of Vectors for Magnetic Field Direction?

    Homework Statement Homework EquationsThe Attempt at a Solution the answer given is the same but without the negative sign, I don't understand because the crossproduct of unit vectors when using a Cartesian coordinates of the directions given by the right-hand rule? Is the positive z...
  26. Michelle027

    Find average acceleration - vectors

    1. A car is traveling 9m/s Northwest. 8 seconds later it has rounded a corner and is now heading North at 15m/s. This was a question from my textbook and was an example question - so they supplied answers. I was able to work through all of it finding everything except the last question. They...
  27. I

    I Convergence Vectors Calculus: Definition

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  28. bwest121

    I Question about Cross Product and Derivative

    Hi everyone, Given a vector-valued function ##\vec{A}##, how do I show that: $$\vec{\nabla} \times \left(\frac{\partial \vec{A}}{\partial x}\right) = \frac{\partial}{\partial x}(\vec{\nabla} \times \vec{A})$$ In other words, are the cross product and derivative commutative w/ each other? I...
  29. davidge

    I Vectors and isometries on a manifold

    Hi. I've been thinking about vectors, coordinate systems and all things associated for a long time. I'd like to know if (at least in the context of General Relativity) my interpretation of these subjects is correct. I will try to summarize my thoughts as follows: - We start with a general...
  30. sampsont

    I Angle between two vectors with many dimensions

    The angle between two ℝ2 or ℝ3 vectors makes sense. I have a vector like [0 0.707 1 0.707 0 -.707 -1 -.707]T (Actually my vector is A/D conversions of a sine wave from the wall power outlet. The sample rate is 4800 so there are 80 samples, separated by 1/Fs for a 60Hz sine wave. If I...
  31. Mr Davis 97

    I DIfference between vectors and relations of points

    When it comes to analytic geometry, I am little confused about the use of vectors. For example, throughout high school, one works in ##\mathbb{R}^2##, and geometric objects such as lines are described using equations relating two variables, the x and y coordinate, such as y = 2x + 1. However...
  32. Z

    Is my textbook wrong about velocity/displacement vectors?

    My textbook states that "The velocity has the same direction as the displacement". I feel this statement is incorrect. Keeping in one dimension, let's say that I move in the +ve direction at velocity v for some time t. My displacement is vt which points in the +ve direction. However I then stop...
  33. O

    A The meaning of an integral of a one-form

    So I understand that the integral of a differential form ω over the boundary of some orientable manifold Ω is equal to the integral of its exterior derivative dω over the whole of Ω. And I understand that one can pull back the integral of a 1-form over a line to the line integral between the...
  34. G

    Splitting reaction into horizontal and vertical components

    Homework Statement In each of the diagrams (please see attached file (I am sorry for the rotated, the original was in normal form but when uploading it, it was somehow rotated)) and the description of each case below, a particle is moving on a smooth surface, so that the reaction force R acting...
  35. A

    Vectors: Displacement, average velocity & speed

    Homework Statement A plane flies 413 km east from city A to city B in 49.0 min and then 814 km south from city B to city C in 1.70 h. For the total trip, what are the (a) magnitude and (b) direction of the plane's displacement, the (c) magnitude and (d) direction of its average velocity, and...
  36. Vitani11

    Are two vectors that are orthogonal to a third parallel?

    Homework Statement Is it true in three dimensions that any two vectors perpendicular to a third one are parallel to each other? Homework Equations Dot product. The Attempt at a Solution I've come up with two vectors that were orthogonal to a third and found the angle between them using the...
  37. karush

    MHB 231.12.3.19 angle between vectors

    $\tiny{231.12.3.19}$ $\textsf{Given $v=-7i-j$ and $w=-i-7j$}\\$ $\textsf{find the angle between v and w}\\$ $\displaystyle \frac{\left(-7, -1, 0\right)\cdot\left(-1, -7, 0\right)} {\sqrt{50}\cdot \sqrt{50}} =\frac{14}{50}\approx 0.28$ $\arccos(0.28)\approx 73.74^o$ $\textit{not sure if this is...
  38. Alettix

    Point inside a tetrahedron with vectors

    Homework Statement As part of a longer problem: "Find necessary and sufficient conditions for the point with positionvector r to lie inside, or on, the tetrahedron formed by the vertices 0, a, b and c." Homework Equations I am not sure... vector addtion? The Attempt at a Solution I don't...
  39. M

    Proving Linear Independence of Av1, ..., Avk and Conditions for Basis in Rm

    Homework Statement [/B] 1. Suppose {v1, . . . , vk} is a linearly independent set of vectors in Rn and suppose A is an m × n matrix such that Nul A = {0}. (a) Prove that {Av1, . . . , Avk} is linearly independent. (b) Suppose that {v1, . . . , vk} is actually a basis for Rn. Under what...
  40. M

    A Why are open strings vectors or scalars, or massive?

    In string theory, if we have NN BCs along ##X^i, i = 1, \ldots, n-1## and DD BCs along ##X^a, a = n, \ldots, 25## then you get, from ##\alpha^{i,a}_{-1}|0,p\rangle ##, ##n## massless vectors and ##24-n## massless scalars. I understand that for the first excited level, ##M^2=0## and so we have...
  41. J

    A How do I supply arpack drivers with all starting vectors?

    I am using arpack (the dsdrv1 driver) to iteratively solve the eigenvalue problem Ax = λx I am interested in the first m eigenvectors, and I have very good initial approximations for these vectors, so I would like to use my m starting vectors as an initial guess. However...
  42. karush

    MHB S6.12.3.17 Find the angle between the vectors

    $\tiny{s6.12.3.17}\\$ $\textsf{ Find the angle between the vectors $a$ and $b$}\\$ \begin{align} \displaystyle a&=\langle 3,-1,5\rangle &b&=\langle -2,4,3\rangle\\ \\ \cos\left({\theta}\right)&=\frac{(3\cdot -2)+(-1\cdot4)+(5\cdot3)} {|\sqrt{35}|\cdot|\sqrt{29}|}\\ &\approx 0.1569 \\...
  43. karush

    MHB S6.12.3.35 Find the unit vectors

    $\tiny{s6.12.3.35}\\$ 35. Find the unit vectors that are parallel to the tangent line to the parabola $y = x^2$ at the point $(2,4)$. \begin{align} \displaystyle y'&=2x \end{align} the book answer to this is $\pm\left(i+4j)/\sqrt{17}\right)$ but don't see how they got this?
  44. M

    MHB Determine the vectors of components

    Hey! :o For the polynomial vector space $\mathbb{R}[x]$ of degree $\leq 3$ we have the following three bases: $$B_1 = \{1 - X^2 + X^3, X - X^2, 1 - X + X^2, 1 - X\} , \\ B_2 = \{1 - X^3, 1 - X^2, 1 - X, 1 + X^2 - X^3\}, \\ B_3 = \{1, X, X^2, X^3\}$$ How can we determine the following...
  45. M

    MHB How to Extend These Vectors to a Basis in $\mathbb{R}^4$?

    Hey! :o Let $t\in \mathbb{R}$ and the vectors $$v_1=\begin{pmatrix} 0\\ 1\\ -1\\ 1 \end{pmatrix}, v_2=\begin{pmatrix} t\\ 2\\ 0\\ 1 \end{pmatrix}, v_3=\begin{pmatrix} 2\\ 2\\ 2\\ 0 \end{pmatrix}$$ in $\mathbb{R}^4$. I want to determine a maximal linearly independent subset of $\{v_1...
  46. doktorwho

    Determining the velocity function

    Homework Statement Given the ## r(t) = ae^{kt}## , ##θ(t)=kt## find the velocity function that is dependent on ##r##. ##v(r)=?## Homework Equations 3. The Attempt at a Solution [/B] My attempt: 1)##r(t) = ae^{kt}## 2)##{\dot r(t)} = ake^{kt}## From the first equation: ##\ln...
  47. C

    Rate of Change of Vector in Rotating Frame

    I recognize the rate of change of a vector in an inertial frame S can be related to the rate of change of the vector in a rotating frame S0 by the equation below taken from my textbook, where Ω is the angular velocity vector. $$\Big(\frac{dQ}{dt}\Big)_{S_{0}}= \Big(\frac{dQ}{dt}\Big)_{S} +...
  48. V

    B Compute Angle Between Multiple 3d Vectors

    Imagine a point with n vectors (all with equal magnitude) coming out from that point that equally cancel each other out in magnitude. How would you calculate the equal angle between n vectors? For example: 2 vectors (equal magnitudes) coming from one point that cancel each others magnitude...
  49. R

    Intro to Angled Forces and Vectors question?

    Homework Statement The 5.7 N weight is in equilibrium under the influence of the three forces acting on it. The F force acts from above on the left at an angle of α with the horizontal. The 5.2 N force acts from above on the right at an angle of 63◦ with the horizontal. The force 5.7 N acts...
  50. Kevin McHugh

    I Wedge product of basis vectors

    Is there a set of relationships for the wedge product of basis vectors as there are for the dot product and the cross product? i.e. e1*e1 = 1 e1*e2 = 0 e1 x e2 = e3
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