What is Vector components: Definition and 98 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. F

    How to define vectors in spherical coordinate system?

    I am extremely confused with how to represent vectors that do not start at the origin in spherical coordinate system. If they did start at the origin, the vector to any point is simply ##r\pmb{\hat{r}}##; however, what if it does not start at the origin as in the question above? One thing I can...
  2. brotherbobby

    To find the component of a vector perpendicular to another

    Problem statement : I copy and paste the (slightly different) problem statement as it appeared in the text to the right. Attempt : By inspection, we find that the vector ##\vec B'## perpendicular to ##\vec B = 3\hat i+4\hat j## is ##\boldsymbol{\vec B' = 4\hat i -3\hat j}##, remembering that...
  3. E

    Vector is a function of its position or not?

    At first I thought that this force vector ## \vec F = 3 \hat x + 2 \hat y ## is a function of ## x ## and ## y ##, which is to say that its magnitude and direction vary with the x and y positions, but this is not so, right? It's just a force with a constant magnitude and direction. And I can...
  4. 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...
  5. redtree

    I Does Each Component of a Vector Have an Independent Fourier Transform?

    Given ##f(\vec{x})##, where the Fourier transform ##\mathcal{F}(f(\vec{x}))= \hat{f}(\vec{k})##. Given ##\vec{x}=[x_1,x_2,x_3]## and ##\vec{k}=[k_1,k_2,k_3]##, is the following true? \begin{equation} \begin{split} \mathcal{F}(f(x_1))&= \hat{f}(k_1) \\ \mathcal{F}(f(x_2))&= \hat{f}(k_2) \\...
  6. I

    Question about vector components

    The answer is D (60 degrees) and I understand how to get that answer. But this assumes that the new velocity's component of v/4 can form right angles with another component of the new velocity. So I'm confused whether vector components always form right angles to each other. When I searched...
  7. K

    Conversion between vector components in different coordinate systems

    I am not completely sure what the formulas ##v_j = v^a\frac {\partial x^j} {\partial \chi^a}## and ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## mean. Is ##v_j## the j:th cartesian component of the vector ##\vec v## or could it hold for other bases as well? What does the second equation...
  8. K

    I Transformation of vector components

    The components of a vector ##v## are related in two coordinate systems via ##v'^\mu = \frac{\partial x'^\mu}{\partial x^\sigma}v^\sigma##. When evaluating this at a specific ##x'(x_0) \equiv x'_0##, how should we proceed? ##v'^\mu(x'_0) = \frac{\partial x'^\mu}{\partial...
  9. J

    Inelastic 2D Collision with Vector Components

    Homework Statement Two balls with mass m and 4m collide at the location x=y=0 and stick. Their initial velocities just before the collision can be represented as v1=(i+j) v and v2=(j-i)v' respectively. Their final velocity vf makes an angle θ with the +x axis. Find v and v' in terms of vf and...
  10. saadhusayn

    I Matrix for transforming vector components under rotation

    Say we have a matrix L that maps vector components from an unprimed basis to a rotated primed basis according to the rule x'_{i} = L_{ij} x_{j}. x'_i is the ith component in the primed basis and x_{j} the j th component in the original unprimed basis. Now x'_{i} = \overline{e}'_i. \overline{x} =...
  11. R

    Calculating Vector Components with given Magnitude and Direction

    Homework Statement F =(70N, 57.1∘counterclockwise from positive y−axis)[/B] Find the vector components of F Homework Equations Sin and Cos of the angle[/B]The Attempt at a Solution x is component is 38 and y component is 58 how does the angle being counterclockwise affect my answer? The...
  12. W

    Methods in Calculating with Vector Components

    <Moderator's note: Moved from a technical forum and therefore no template.> Homework Statement A proton (q = 1.60 x 10-19) is in a uniform, 0.500 T magnetic field. This proton has velocity components vx = 1.50 x 105, vy = 0, and vz = 2.00 x 105 m/s. Find the force on the proton at t=0. 2...
  13. C

    Calculating Work with Vector Components and Variable Force

    Hello all! I usually don't like to ask for help... But this is the first week of courses and I'm already stumped on a homework question... 1. Homework Statement So the question states: Find the work by the force F = x^i + xy^j. If the object starts from the origin (0,0), moves along the...
  14. saadhusayn

    I Transformation of covariant vector components

    Riley Hobson and Bence define covariant and contravariant bases in the following fashion for a position vector $$\textbf{r}(u_1, u_2, u_3)$$: $$\textbf{e}_i = \frac{\partial \textbf{r}}{\partial u^{i}} $$ And $$ \textbf{e}^i = \nabla u^{i} $$ In the primed...
  15. C

    MHB Finding X & Y Components of Vectors

    Im learning about how to find the x and y components of a vector, but I wanted to verify if I'm on the right track.. Ax=A cos(\theta) <-- solving for x Ay=A sin(\theta) <-- solving for y So if a vector is 9.55m long and points in a -48.0 degree direction. Is it Ay= 9.55 sin(-48.0)=7.3
  16. P

    I L due to Azimuthal (1/r)x(1/r^2) Poynting vector components?

    The basic idea: I am interested in the possibility of an azimuthally-directed Poynting vector component which drops with the inverse cube of the distance (or as 1/r^3), primarily because it suggests the possibility of emitting field angular momentum, allowing for a uni-directional torque to be...
  17. redtree

    I Fourier transform of the components of a vector

    Given the Fourier conjugates ##\vec{r}## and ##\vec{k}## where ##\vec{r} = [r_1,r_2,r_3]## and ##\vec{k} = [k_1,k_2,k_3]## , are ##r_1## /##k_1##, ##r_2##/##k_2##, ##r_3##/##k_3## also Fourier conjugates, such that: ##\begin{equation} \begin{split} f(\vec{r})&=[f_1(r_1),f_2(r_2),f_3(r_3)] \\...
  18. F

    I Vector components, scalars & coordinate independence

    This question really pertains to motivating why vectors have components whereas scalar functions do not, and why the components of a given vector transform under a coordinate transformation/ change of basis, while scalar functions transform trivially (i.e. ##\phi'(x')=\phi(x)##). In my more...
  19. A

    Finding Magnitudes of Forces in Similar Triangles

    An upward force of magnitude 30N is combined with a second force F which makes an angle of 10degree with the horizontal. The resultant makes an anglre of 40degree with the horizontal. a. Determine the magnitude of the second force b. Determine the magnitude of the resultant
  20. Square1

    Adding vector components in two dimensions

    Homework Statement A plane flies 600km/h south. The plane encounters a southwestern wind of 100km/h. Homework Equations What is the velocity (magnitude and direction) of the plane The Attempt at a Solution What I did was break the 100km southwestern vector into its components. 100*cos45 =...
  21. S

    I Original direction of force versus vector components

    What happens to a mechanical force's real original direction i.e. when we divide it into components of basis vectors, which in turn change as per problem at hand (like gravity components at inclined plane ), how we arrive at correct physics by taking two/three arbitrary directions of our choice...
  22. tomdodd4598

    Can the Dot Product of Four Vectors Maintain Positive Component Signs?

    Hi there, I understand that taking the dot product of two four vectors automatically applies the metric tensor to the second vector. Is there a way to take write the dot product, using vector notation, in a way which keeps the signs of all of the components positive? Thanks in advance.
  23. M

    I want to find the vector components of an MRI pixel -- How?

    I have a Dicom Image that is interpreted as an array. Each cell is a pixel. Each pixel has a value. I want to find the vector components (x,y,z) of a pixel. Specifically, I want to find magnetic field vectors for each pixel. The image is for H1 atoms. Freq = 297, B0 = 3. How would you do this...
  24. J

    General Lorentz transformation, spatial vector components

    Hello, a derivation of the lorentz transformation for an arbitrary direction of the relative velocity often makes use of writing the spatial position vector of an event as the sum of its component parallel and perpendicular to the velocity vector in one inertial frame and then transforming both...
  25. Brendan Webb

    Finding the Magnitude and Direction of the Displacement Vector for a Ship's Trip

    A ship sails 130 km due north from island A to island B and then 92 km, in the direction 22° south of east, to island C. The ship after that returns directly to island A. Calculate the magnitude and direction of the displacement vector in the last trip. Draw appropriate diagrams. Homework...
  26. upwardfalling

    Adding Vectors Using the Component Method

    Homework Statement Use the component method to add the vectors vector A and vector B shown in the figure. The length of vector B is 3.25 m and the angle θ = 28.5°. Express the resultant vector A + vector B in unit-vector notation. Homework Equations x = rcos y = rsin The Attempt at...
  27. P

    Vector components in polar coordinates

    The magnitude of the parallel component of the time derivative of a vector ##\vec{A}## is given by: $$|\frac{d\vec{A}_{\parallel}}{dt}| = |\frac{dA}{dt}|$$ Where ##A## is the magnitude of the vector. Can we write the actual derivative in vector form as ##\frac{dA}{dt} \hat{A}##? Notice how I...
  28. M

    Projectile motion- solve for initial velocity

    Homework Statement A ball is thrown upward at an angle of 30degrees to the horizontal and lands on the top edge of a building that is 20m away. The top edge is 5.0m above the throwing point. How fast was the ball thrown? Homework Equations x=v0t+(1/2)at2 vf2=V02+2ax vx=v0cosθ vy=v0sinθ The...
  29. S

    Determining Angle with Vector Components

    Greetings women and men, I have a problem in which I have to find an angle \phi. http://srg.sdf.org/images/PF/StaticsHW.png A horizontal force of \vec{F}=400 lbs is placed on the structure at point A. Find angle \phi to give the AB component of \vec{F} a magnitude of 600 lbs. To solve this...
  30. K

    Determining Displacement Using Vector Components

    Homework Statement This problem has had me puzzled for hours. I really appreciate any help you can provide. A light plane leaves Shelburne, NS and flies 195km [N15°W] to Saint John, NB. After picking up a passenger, it flies to Moncton, NB which is 149km [N33°E]. The entire trip took...
  31. F

    Understanding Perpendicular Vector Components

    "Is it possible for a vector that has nonzero magnitude to have a component in some direction that is equal to zero?" The answer key said that any vector that has a nonzero magnitude will always have a component of zero length in the direction perpendicular to the vector. I'm having trouble...
  32. S

    How Is Average Velocity Calculated in Multi-Stage Vector Displacements?

    You go for a short walk traveling in three stages. The first displacement is 58.5 m 20.0 degrees east of north. The second displacement is 78.0 m 40.0 degrees south of east. Finally you go 99.0 m 17.0 degrees north of west. The answer I got was 36.9 m, 66.2 degrees north of west,which is...
  33. N

    Projection of vector vs vector components

    Homework Statement This is an example problem where you have a force F at 100N applied at an angle of 45 degrees from a horizontal u-axis. You have the u-axis at zero degrees, then 45 degrees after that you have the Force then 15 degrees after th at you have the v-axis You are asked to...
  34. E

    Calculating vector components (displacement, oooh)

    Homework Statement Sophie walks 800 M [N], then 500 M [W] and finally 400 M [SE] in 25 minutes. What is her displacement and average velocity? (Using scale diagram) Homework Equations a2 + b2 = c2 The Attempt at a Solution I tried calculating by creating separate triangles, but...
  35. D

    Vector components and its coordinate description in a given basis

    Given a basis \mathfrak{B}=\lbrace\mathbf{e}_{i}\rbrace it is possible to represent a vector \mathbf{v} as a column vector \left[\mathbf{v}\right]_{\mathfrak{B}}= \left(\begin{matrix}v^{1} \\ v^{2} \\ \vdots \\ v^{n}\end{matrix}\right) where the v_{i} are the components of \mathbf{v} relative...
  36. A

    Unit Vectors and Vector Components

    I remember in my first year physics classes, when dealing with a force F we would find the vector's x- and y-components using F_x=r cos(θ) and F_y=r sin(θ) I also remember learning in my mathematics classes about unit vectors, but cannot seem to remember using them to break vectors down into...
  37. J

    What are the vector components of a rocket's impact velocity?

    A rocket hits the ground at an angel of 60° from the horizontal at a speed of 300 m/s. a. Draw the vector representing the rocket's impact and show the westward and eastward components of it's velocity. b. Calculate the horizontal and vertical components of the rocket's impact velocity.
  38. J

    How Do You Calculate Vector Components of a Rocket's Velocity?

    A rocket heads at an angel of 40° West of North at a speed of 150 m/s. A Draw the vector representing the planes flight and show the westward and northward components of it's velocity. b. Calculate the westward and northward components of the plane's velocity. Please i really need help...
  39. C

    How Do You Calculate the Effective Speed of a Tailwind for a Westward Cyclist?

    Homework Statement A cyclist head due west on a straight road at 5.6m/s. A northeast wind is blowing at 10m/s. What is the effective speed of the tailwind?(resultant) Homework Equations Cos 45 10 Sin 45 10 The Attempt at a Solution Basically I broke it down into its x and y...
  40. C

    Vector Components: Equal Force of Two Wires Suspending 500N Sign

    Homework Statement two wires are used to suspend a sign that weighs 500N. the two wires make an angle of 100 degrees between each other. If each wire is exerting an equal amount of force, how much force does each wire exert? Homework Equations The Attempt at a Solution The answer...
  41. P

    Vector Components and Translation

    Inspired by a question in Griffiths' E&M book (1.10), I am wondering why the components of a vector do not change when the coordinate system is translated by a constant vector. I understand that, for instance, the velocity of something moving in a coordinate system won't change if we then...
  42. P

    Problem pertaining to relative velocity and vector components

    A woman who can row a boat at 6.4 km/h in still water faces a long, straight river with a width of 6.3 km and a current of 3.3 km/h. Let i point directly across the river and j point directly downstream. If she rows in a straight line to a point directly opposite her starting position, (a)...
  43. B

    Vector Components and Dot Product Proof

    Homework Statement A is a vector Show that: a1 = x hat (dot) A a2 = y hat (dot) A a3 = z hat (dot)A Homework Equations A= (a1*x hat) + a2*y hat) + (a3* z hat) The Attempt at a Solution my hint says to take the dot product of...
  44. N

    Resolving vector components using trigonometry

    Homework Statement Find the components of the vector A with length a = 1.00 and angle=20.0° with respect to the x-axis as shown. Enter the x component followed by the y component, separated by a comma. Homework Equations none The Attempt at a Solution To find the X-component...
  45. C

    Kinematics question projectiles with vector components (baseball player Q)

    Homework Statement Base ball player makes perfect contact with a ball striking it at 45° angle above the horizontal at a point 1.3 m above the ground. His ball just makes it clear of the 3m wall 130m from home plate. What was the velocity at with he struck the ball. Homework Equations is...
  46. L

    Vector components on Tilted Axis

    θHomework Statement Find the components N_{x} and N_{y} of vector N in the tilted cooridinate system. Homework Equations cosθ = adg/hyp and sinθ= opp/hyp The Attempt at a Solution The correct answer is supposed to be N_{x} = -Nsinθ and N_{y} = Ncosθ The error...
  47. V

    Calculating Resultant Components

    What are the resultant components of the vectors? Homework Statement Determine the sum of the following three vectors. Give the resultant in terms of a. components, b. magnitude and angle with x-axis. Vector A has a magnitude of 66 at an angle of 28 degrees northeast (quadrant 1) Vector...
  48. N

    Vector Components and Displacement of Lost Dog

    Homework Statement A lost dog travels 30 degrees south of west for 450m. He ten smells some food and sprints at 20 degrees east of north for 600m. a) draw each vector. Label magnitude and theta. b) calculate the vector components c) calculate the magnitude of the dog's displacement...
  49. E

    Vector Components for Destination Distance

    Homework Statement Amy wants to reach a destination that is 330m [E31 degrees N] from where she is. She travels a distance b towards [E20 degrees N] and then 100m to her destination. What is the distance b? (list all possible answers) (Sorry I don't know how to get symbols) Let [N] and [E] be...
  50. J

    Calculating Vector Components: Finding B_y with A-B Parallel to -x Axis

    Homework Statement The diagram below shows two vectors, A and B, and their angles relative to the coordinate axes as indicated. http://loncapa.physics.mun.ca/res/mun/PHYSICS/msuphysicslib/Graphics/Gtype07/prob01a_vectors2.gif DATA: α=43.7°; β=53.4°; |A|=4.30 cm. The vector A−B is...