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. Euler2718

    Linear Dependence and Non-Zero Coefficients

    Homework Statement True or False: If u, v, and w are linearly dependent, then au+bv+cw=0 implies at least one of the coefficients a, b, c is not zero Homework Equations Definition of Linear Dependence: Vectors are linearly dependent if they are not linearly independent; that is there is an...
  2. Erenjaeger

    Finding orthogonal unit vector to a plane

    Homework Statement find the vector in R3 that is a unit vector that is normal to the plane with the general equation x − y + √2z=5 [/B]Homework EquationsThe Attempt at a Solution so the orthogonal vector, I just took the coefficients of the general equation, giving (1, -1, √2)[/B] then...
  3. M

    MHB Can the vectors be written as a linear combination?

    Hey! :o We have the vectors $\overrightarrow{a_1}=\begin{pmatrix}1 \\ 2 \\ 3\end{pmatrix}, \overrightarrow{a_2}=\begin{pmatrix}-1 \\0 \\ 2\end{pmatrix}, \overrightarrow{a_3}=\begin{pmatrix}7 \\ 8 \\ 6\end{pmatrix}$. I have shown that these vectors are linearly dependent: $\begin{bmatrix}...
  4. Matejxx1

    Vector algebra (proving you have a parallelogram by using vectors)

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  5. A

    I Is the Dot Product of Unit Vectors Related to Magnitudes and Angle Between Them?

    Okay so I understand that in order to represent a vector which is in cartesian coordinates in spherical coordinates.. we use the transformation which is obtained by dotting the unit vectors. So my question goes like this: when we dot for example the unit vector ar^ with x^ we obtain sin(theta)...
  6. E

    I Order of Operations for Tensors

    Hey so probably a really simple question, but I'm stumped. How do you simplify: ν∇⋅(ρν), where ν is a vector ∇ is the "del operator" ⋅ indicates a dot product ρ is a constant. I want to say to do the dyadic product of v and ∇, but then you would get (v_x)*(d/dx) + ... which would be...
  7. Mr Davis 97

    I Representing vectors with respect to a basis

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  8. G

    I Prove ∇(A x B) = (∇ x A)⋅B - (∇ x B)⋅A where A,B are vectors

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  9. S

    Let A, B, and C be vectors in R3....prove r lies in a plane

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  10. R

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  11. Iron_Man_123

    Can you me answer this question about vectors

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  12. G

    I Differentiating Vectors and their Magnitudes in Physics

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  13. M

    What Is the Dimension of Eigenspaces for Given Characteristic Polynomial?

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  14. S

    What did I do wrong? Two-Dimension Vectors

    1. Homework Statement So the problem is: A woman walks 440 m at 50° S of W and then 580 m at 60° N of E. The entire trip required 15 minutes. A. What was the total distance (I already got the answer to this) B. What was the displacement of the woman?Homework Equations I'm almost positive I did...
  15. D

    Calculate the electric field using superposition

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  16. T

    I How are Vectors described in Bispherical Coordinates?

    I was reading a paper that described a vector field in terms of its three components , ##A_σ,A_τ,A_φ##. with σ, τ and φ being the three bispherical coordinates. what does ##A_σ## mean? In what direction does the component point? Likewise for the other two components.
  17. FallenApple

    Rotational Vectors not merely a bookkeeping device?

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  18. A

    I Plotting the orbits of the planets

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  19. L

    Ball trajectory velocity and displacement vectors

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  20. R

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  21. R

    Solving Cartesian Vectors in Mechanics: Expressing Force F(AB) in Cartesian Form

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  22. S

    Vectors and covectors under change of coordinates

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  23. R

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  24. R

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  25. M

    I Expanding linear independent vectors

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  26. P

    Polar coordinates of a vector

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  27. terhje

    Vectors: check if coordinates are in the same plane

    Hello guys, How can i check if coordinantes A,B,C and D are in the same plane? 3D space(x,y,z) Can i take the cross product: AB x AC and check if its perpendicular to for example DC x DB. and then check if the crossproducts are parallell? but i guess this can give me two parallell vectors in...
  28. steele1

    Prove area of triangle is given by cross products of the vertex vectors....

    Homework Statement The three vectors A, B, and C point from the origin O to the three corners of a triangle. Show that the area of the triangle is given by 1/2|(BxC)+(CxA)+(AxB)|. Homework EquationsThe Attempt at a Solution I know that the magnitude of the cross product of any two vectors...
  29. Vanessa Avila

    How to solve for the magnitude and angle in this situation?

    Homework Statement Each one of the force vectors are surrounded with dimensions. But you're not given any angles. How do you solve for the resultant vector like that? Homework Equations Fr = F1 + F2 + F3 The Attempt at a Solution I tried taking the arctan(200/210) on the 435 N vector to...
  30. Hackerjack

    Vector problem missing magnitude of two vectors

    Homework Statement Force A has a magnitude of 200 lb and points 35deg N of W. Force B points 40deg E of N. Force C points 30deg W of S. The resultant of the three forces has a magnitude of 260 lb and points 85deg S of W. Find the magnitude of forces B and C. Homework Equations...
  31. Mr Davis 97

    B Vector perpendicular to a plane defined by two vectors

    Say that I have two vectors that define a plane. How do I show that a third vector is perpendicular to this plane? Do I use the cross product somehow?
  32. Mr Davis 97

    I Show that the diagonals are perpendicular using vectors

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  33. F

    Trying to understand vector conversion matrices

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  34. Mr Davis 97

    I Difference between vectors in physics and abstract vectors

    I am taking a linear algebra course and an introductory physics course simultaneously, so I am curious about the connections between the two when it comes to vectors. In beginning linear algebra, you typically study vectors in ## \Re^{2}## and ## \Re^{3}##. Are these the same vector spaces used...
  35. Mr Davis 97

    B The equation relating a vector to a unit vector

    I am studying physics, and I see the equation ##\hat{A} = \frac{\vec{A}}{A}##. What makes this relation obvious? It's quite obvious when one of the components of vector A is zero, but if both components are not zero, then what leads me to believe that this relation works every time?
  36. RJLiberator

    Coordinate representation of vectors?

    Homework Statement Starting from the coordinate representation for the vectors, show the result in Equation 1.16 of Griffith's book. (1.16)A \cdot (B \times C) = \left[ \begin{array}{ccc} A_x & A_y & A_z \\ B_x & B_y & B_z \\ C_x & C_y & C_z \end{array} \right] Note: Here, I use * to...
  37. D

    I The two equivalent parallel velocity vectors

    This is an exercise from the textbook Apostol Vol 1 (page 525, second edition), and I do not know how to prove it: Suppose a curve C is described by two equivalent functions X and Y, where Y(t) = X[u(t)]. Prove that at each point of C the velocity vectors associated with X and Y are parallel...
  38. Pao44445

    Calculating Resultant Vector with Given Angle and Magnitudes?

    Homework Statement find the scale of R and angle Homework Equations Vector A = 100 Vector B = 200 The Attempt at a Solution I know that I need to find the answer of this equation "R = A+B" but I can't find Bx and By because of the angle
  39. gracy

    Resultant of two vectors of equal magnitude

    Homework Statement Resultant of two vectors of equal magnitude A is a) √3 A at 60 b) √2 A at 90 c) 2A at 120 d) A at 180 Homework Equations When two vectors are of equal magnitudes then their resultant is ##A_R## = 2 A Cos θ/2 The Attempt at a Solution I think we need more information...
  40. S

    I Linearizing vectors using Taylor Series

    I am linearizing a vector equation using the first order taylor series expansion. I would like to linearize the equation with respect to both the magnitude of the vector and the direction of the vector. Does that mean I will have to treat it as a Taylor expansion about two variables...
  41. peasqueeze

    I Use Lorentz Force to Find Magnetic Field Components

    So I am constructing an analogy between the self replicating fracturing effect on thin films and the path of a charged particle. (Qualitatively, several cracks have similar shapes to charged particle motion) I won't go into the details of the fracture mechanics, so I will only use E+M...
  42. M

    Calculating Linear Combinations of Vectors: Step-by-Step Guide

    Homework Statement Consider the four vectors (1, 1, 1), (2, −1, 3), (1, 7, −1) and (1, 4, 0). Calculate how many ways you can write (1, 1, 1) as a linear combination of the other three, explaining your reasoning. The Attempt at a Solution wouldn't any of these combinations give the correct...
  43. D

    Determining the Max. Set of Linearly Independent Vectors

    (sorry for the horrible butchered thread title... should say "determination", not "determining") 1. Homework Statement In "Principles of Quantum Mechanics", by R. Shankar, a vector space is defined as having dimension n if it can accommodate a maximum of n linearly independent vectors (here is...
  44. RoboNerd

    I Finding shortest distance between skew lines, checking work.

    Hi everyone. I was working on a problem for days. The problem statement is: "Consider points P(2,1,3), Q(1,2,1), R(-1,-1,-2), S(1,-4,0). Find the shortest distance between lines PQ and RS." Now, I did the following formula: PS dot (PQ x RS) / magnitude of (PQ x RS). (For skew lines) Now...
  45. D

    How Can I Find Eigenvalues and Normalized Eigenvectors for a Matrix?

    Homework Statement Find the eigen values and normalized eigen vectors for the matrix cosθ sinθ -sinθ cosθ 2. The attempt at a solution Well I did the eigen values hope they are correct but can't solve for eigen vectors Eigen values are λ = cosθ ± isinθ on solving for eigen vector for...
  46. RoboNerd

    Question about determining the angles of triangle given two vectors

    <<Mentor note: Missing template due to originally being posted elsewhere>> Hello everyone. I have the following problem: Determine the angles of a triangle where two sides of a triangle are formed by the vectors A = 3i -4j -k and B=4i -j + 3k I thought that I would find the third side being...
  47. F

    B Explaining vector & scalar quantities to a layman

    I've been asked by someone with minimal background in physics to explain what vector and scalar quantities are and give examples. Here are my thoughts: A scalar is a quantity that has a magnitude only, it is completely specified by a single number. Importantly, it has no directional dependence...
  48. V

    I (2,0) tensor is not a tensor product of two vectors?

    Hi. I'm trying to understand tensors and I've come across this problem: "Show that, in general, a (2, 0) tensor can't be written as a tensor product of two vectors". Well, prior to that sentence, I would have thought it could... Why not?
  49. H

    B Vectors & Gradients: Confused? Get Answers Now!

    I was watching a video explaining Einsteins field equations for beginners and he was giving some information on Vectors before he gets into the actual equations. He got to this equation: I'm real confused, how does he know that dx and dø are both? Why does dø change to dy? What does the...
  50. parshyaa

    I Why do we need projection of vectors

    I know that projection of vector B on A is ||B||cos(theta) where theta is the angle between vector A and B . But why do we find it . Is there any application for this
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