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. H

    Vector Nature pf Projectile Motion

    Homework Statement If an object had been projected horizontally with the same magnitude as in the depicted situation, how would the motion compare with that of the object in the diagram? (I have drawn the diagram in my attachment and have done questions c and d but I don't understand question...
  2. B

    A Spinor Lorentz Transform via Vectors - Cross Product Issue

    The Lorentz transformation operator acting on an undotted, i.e. right-handed, spinor can be expressed as $$e^{-\frac{1}{2} \sigma \cdot \mathbf{\phi} + i\frac{1}{2} \sigma \cdot \mathbf{\theta}}.$$ There is a very cool, almost childlike, derivation of this expression in Landau Vol. 4 S. 18 I've...
  3. H

    Adding Vectors using Cefore Method

    A bird is flying through the air with an air speed of 12.6 m/s heading 27.7° west of north. It is a windy day with the wind blowing at 8.95 m/s heading 12.5° south of west. What is the velocity of the bird as seen by a person on the ground? Note: I keep doing this question but my answer (the...
  4. 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...
  5. 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...
  6. 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}...
  7. Matejxx1

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

    Homework Statement 23. In a ABCD quadrilateral let P,Q,R,S be midpoints of sides AB,BC,CD and DA. Let X be the intersection of BR and DQ, and let Y be the intersection of BS and DP. If ##\vec{BX}=\vec{YD} ## show that ABCD is a parallelogram . Homework Equations ## (\vec{a}\cdot\vec{b})=0##...
  8. 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)...
  9. E

    I How Do You Simplify ν∇⋅(ρν)?

    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...
  10. Mr Davis 97

    I Representing vectors with respect to a basis

    I'm a little bit confused about how coordinate systems work once we have chosen a basis for a vector space. Let's take R^2 for example. It is known that if we write a vector in R^2 numerically, it must always be with respect to some basis. So the vector [1, 2] represents the point (1, 2) in the...
  11. G

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

    I can prove this relationship by defining A = (A1,A2,A3) and B=(B1,B2,B3) and expanding but I tried another approach and failed. I read that for any 3 vectors, a⋅(b x c) = (a x b)⋅c and thus applying this to the equation, I only get ∇(A x B) = (∇ x A)⋅B Can anyone explain why this is so?
  12. S

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

    Homework Statement ##\vec { \dot { r } } =(t+1)\vec { A } +(1-sint)\vec { B } \quad \vec { r(0) } =\vec { C } ## a. Find an equation of the tangent line to the curve at ##\vec { r(0) } =\vec { C } ##. b. Use a definite integral to find ##\vec { r(t) } ## c. If ##A## and ##B## are non...
  13. R

    Force vectors: How to determine the directions of their components

    Homework Statement Replace the force system by an equivalent resultant force and couple moment at point P. Suppose thatF1 = 455 N and F2 = 700 N . (Figure 1) Homework EquationsThe Attempt at a Solution I found the solution to the problem. My question is how do I know which direction the...
  14. Iron_Man_123

    Can you me answer this question about vectors

    Homework Statement Homework Equations N/A The Attempt at a Solution I tried to sketch the paths but i sill ended up blank
  15. G

    I Differentiating Vectors and their Magnitudes in Physics

    I've taken up to Calc. III (vector/multi-variable calculus) but have not had any classes that used intensive calculus for a few semesters. However, I'm now in a quantum physics class (its a glorified title really--it's more or less "modern physics") and am looking to see if I am correct in my...
  16. M

    What Is the Dimension of Eigenspaces for Given Characteristic Polynomial?

    Homework Statement For c not equal to ±1, what is the dimension of the eigenspaces of A The characteristic polynomial of A is (x-1)(x+1)(x-c) The Attempt at a Solution each term in the characteristic polynomial has a multiplicity of 1 so does this mean that the dimension of the eigenspaces...
  17. 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...
  18. D

    Calculate the electric field using superposition

    Homework Statement Find the expression for the electric field at point M(a,a,0) if the linear charge density is known ( ##Q'## ) Homework Equations 3. The Attempt at a Solution [/B] I tried something like this and would like your feedback on it. I separated the triangle into three parts...
  19. 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.
  20. FallenApple

    Rotational Vectors not merely a bookkeeping device?

    I was told that the direction of the cross product is an arbitary convention to give rotation a "direction" + for one direction and - for the other. That it is simply a book keeping device to make sure different rotation directions are given different signs. It seemed to be the case as I am...
  21. A

    I Plotting the orbits of the planets

    Hello everybody! Long-time lurker and second-time posting. I'm working on a project for my math class, and I'm trying to plot the orbits of the planets using vectors. I've chosen to use MATLAB because I am decently familiar with it. I've used the formulas described in this post here to get my...
  22. L

    Ball trajectory velocity and displacement vectors

    Homework Statement ball is thrown at slighta angle upwards and to the right. Angle between horizontal and the launch velocity vector is 35deg upwards. Launch velocity vector (v_0) has magnitude of 12m/s Only gravity affects the ball as a force. (no air resistance) t_0 = 0 secs t_1 = 0.2 secs...
  23. R

    What Is the Angle Between Position Vectors AB and AC?

    Homework Statement FInd the angle theta between AB and AC. a-1.7m, b=1.3m masteringengineering.com says my answer is wrong. I'm not sure of any other way to do the problem. Please advise. Homework Equations A(3, 0, 0) B(0, -.75, 1.3) C(0, 1.7, 1.5) The Attempt at a Solution r(AC)=...
  24. R

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

    Homework Statement Express force F(AB) in cartesian vector form[/B] Mastering engineering.com says to check my signs but I don't an see an issue. Please help! 2. Homework Equations 3. The Attempt at a Solution A(0, -.75, 3) B(2cos40, 2sin40, 0) C(2,-1, 0) r(AB)= 1.53i +2.04j-3k...
  25. S

    Vectors and covectors under change of coordinates

    Homework Statement If ##\bf{v}## is a vector and ##\alpha## is a covector, compute directly in coordinates that ##\sum a_{i}^{V}v^{i}_{V}=\sum a_{i}^{U}v^{j}_{U}##. What happens if ##\bf{w}## is another vector and one considers ##\sum v^{i}w^{i}##? Homework Equations The Attempt at a...
  26. R

    Force Resultant Along x' Axis: Find F & Theta

    Homework Statement Three forces act on the bracket, the resultant force is directed along the x' axis and has a magnitude of 9.8kN. Determine the magnitude of F. Determine the direction theta of F. Homework EquationsThe Attempt at a Solution I'm really not sure how I am doing with. Any...
  27. R

    Vectors Homework: Displacement & Direction at 80°

    Homework Statement You go to an amusement park with your friend Betty, who wants to ride the 90-m-diameter Ferris wheel. She starts the ride at the lowest point of a wheel that, as you face it, rotates counterclockwise. What is the magnitude of her displacement vector when the wheel has...
  28. M

    I Expanding linear independent vectors

    Hi PF! The other day in class my professor mentioned something about expanding linear independent vectors, but he did not elaborate. From what I understand, if ##x_1,...,x_k## are linearly independent vectors in ##V##, where ##dimV=n>k##, how would you extend ##x_1,...x_k## to a basis ##\{...
  29. P

    How Do Polar Coordinates Explain a Bead's Velocity on a Rotating Wheel?

    Note: All bold and underlined variables in this post are base vectors I was reading the book 'Introduction To Mechanics' by Kleppner and Kolenkow and came across an example I don't quite understand. The example is this: a bead is moving along the spoke of a wheel at constant speed u m/s. The...
  30. 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...
  31. 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...
  32. 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...
  33. 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...
  34. 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?
  35. Mr Davis 97

    I Show that the diagonals are perpendicular using vectors

    I am given the following problem: Show, using vectors, that the diagonals of an equilateral parallelogram are perpendicular. First, imagine that the sides of the equilateral parallelogram are the two vectors ##\vec{A}## and ##\vec{B}##. Since the figure is equilateral, their magnitudes must be...
  36. F

    Trying to understand vector conversion matrices

    Homework Statement This isn't exactly a "problem" per se , but I need to understand it for a course I'm taking. I'm trying to understand the significance and when to use the vector conversion matrices, or just the identities. I'll use an example that I made up, using rectangular to polar...
  37. 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...
  38. 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?
  39. 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...
  40. 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...
  41. 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
  42. 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...
  43. 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...
  44. 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...
  45. 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...
  46. 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...
  47. 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...
  48. 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...
  49. 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...
  50. 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...
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