What is Unit vectors: Definition and 140 Discussions

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in



{\displaystyle {\hat {\mathbf {v} }}}
(pronounced "v-hat").The term direction vector is used to describe a unit vector being used to represent spatial direction, and such quantities are commonly denoted as d; 2D spatial directions represented this way are numerically equivalent to points on the unit circle.
The same construct is used to specify spatial directions in 3D, which are equivalent to a point on the unit sphere.

The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e.,







{\displaystyle \mathbf {\hat {u}} ={\frac {\mathbf {u} }{|\mathbf {u} |}}}
where |u| is the norm (or length) of u. The term normalized vector is sometimes used as a synonym for unit vector.
Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination of unit vectors.
By definition, the dot product of two unit vectors in a Euclidean space is a scalar value amounting to the cosine of the smaller subtended angle. In three-dimensional Euclidean space, the cross product of two arbitrary unit vectors is a third vector orthogonal to both of them, whose length is equal to the sine of the smaller subtended angle. The normalized cross product corrects for this varying length, and yields the mutually orthogonal unit vector to the two inputs, applying the right-hand rule to resolve one of two possible directions.

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  1. srnixo

    Help about the unit vectors for polar coordinates in terms of i and j

    "Firstly, I represented [Uθ ]on the two-dimensional polar coordinate system to facilitate the steps and projections." Then, I have written the steps, step by step, to ultimately derive the expression U(θ) in terms of i and j which is: [ Uθ=−sin(θ)i+cos(θ)j ] NOTE: The professor provided us...
  2. TGV320

    I Law of Cosines in Linear Algebra: Understanding the Dot Product of Unit Vectors

    HI, I am studying linear algebra, and I just can't understand why "Unit vectors u and U at angle θ have u multiplied by U=cosθ Why is it like that? Thanks
  3. 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 =...
  4. 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 ?
  5. 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...
  6. 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...
  7. 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...
  8. D

    How Can I Solve Question Type: "With Magnitude and Unit Vectors"?

    Hi I am a beginner in this topic. I didn't understand this question type clearly.What does it mean" With Magnitude and Unit Vectors" exactly? May you help me for the solution step by step :). Thanks in advance.
  9. O

    What's the use of unit vectors?

    Homework Statement: Hallo. Can somebody explain to me what's the importance-use of unit vector in the below (second) equation? Why isn't the first equation just enough to describe r? What's the reason for unit vector to even exist? Homework Equations: in the photos
  10. torito_verdejo

    Advantages of Polar Coordinate System & Rotating Unit Vectors

    What is the advantage of using a polar coordinate system with rotating unit vectors? Kleppner's and Kolenkow's An Introduction to Mechanics states that base vectors ##\mathbf{ \hat{r}}## and ##\mathbf{\hat{\theta}}## have a variable direction, such that for a Cartesian coordinates system's base...
  11. F

    I Polar coordinates and unit vectors

    Hello, I get that both polar unit vectors, ##\hat{r}## and ##\hat{\theta}##, are unit vectors whose directions varies from point to point in the plane. In polar coordinates, the location of an arbitrary point ##P## on the plane is solely given in terms of one of the unit vector, the vector...
  12. Mason Smith

    Cylindrical coordinates: unit vectors and time derivatives

    Homework Statement Homework EquationsThe Attempt at a Solution I have found expressions for the unit vectors for cylindrical coordinates in terms of unit vectors in rectangular coordinates. I have also found the time derivatives of the unit vectors in cylindrical coordinates. However, I am...
  13. Zack K

    Location of charged particle given magnitude of position

    Homework Statement A charged particle has an electric field at ##\langle -0.13, 0.14, 0 \rangle## m is ##\langle 6.48\times10^3, -8.64\times10^3, 0 \rangle## N/C. The charged particle is -3nC. Where is the particle located? Homework Equations ##\vec E=\frac 1 {4π\varepsilon_0} \frac q {|\vec...
  14. sams

    I A Question about Unit Vectors of Cylindrical Coordinates

    I wrote the equations of the Nabla, the divergence, the curl, and the Laplacian operators in cylindrical coordinates ##(ρ,φ,z)##. I was wondering how to define the direction of the unit vector ##\hat{φ}##. Can we obtain ##\hat{φ}## by evaluating the cross-product of ##\hat{ρ}## and ##\hat{z}##...
  15. T

    Unit Vectors and Momentum Changes in a Block of Ice

    Homework Statement A 0.5 kg block of ice is sliding by you on a very slippery floor at 2.5 m/s. As it goes by, you give it a kick perpendicular to its path. Your foot is in contact with the ice block for 0.0035 seconds. The block eventually slides at an angle of 24 degrees from its original...
  16. K

    Cartesian unit vectors in terms of cylindrical vectors

    How do I express ex,ey,ez in terms er,eθ,eZ? r=(x^2+y^2)^1/2,θ=arctan(y/x),Z=z A(r,θ,z) ∂A/∂x=x/(x^2+y^2)^1/2er+(-y)/(x^2+y^2)eθ=cosθer-(sinθ/r)eθ ex=(∂A/∂x)/|∂A/∂x| I should get ex as cosθer-sinθeθ, but I don't get ex correctly. am i doing this wrong?
  17. Deadawake

    Calculate Change in Speed Using Unit Vectors: Easy Physics Solution

    Actually that's very easy question but I have some difficult to understand the logic behind . So-"The initial velocity of an object (m/s) is Vi=1i+5j+2k. And the final velocity is Vf=3i+5j+7k. What was the change in speed of the object?"X Solution - |Vf|-|Vi| = √(32+52+72)-√(12+52+22) = 3.63...
  18. 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.
  19. B

    Statics: Dimensionless Unit Vector

    Homework Statement Homework Equations The Attempt at a Solution So I began by subtracting. (205-160)=55 i (495+128)=623 j Both of these vectors are in the positive direction. So if I divide the vector by its magnitude I should get an answer of 1 in the positive direction for both i and...
  20. 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?
  21. 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)...
  22. Y

    Unit Vectors as a Function of Time?

    Homework Statement Say I have a vector F something like F = c1(t) x^ + c2(t) y^ were c1 and c2 are some scalar functions of time were you plug in time to into the equation and are given some magnitude. My question seems to be can we define unit vectors/basis vector as a function of time as...
  23. PhysicsKid0123

    Time Derivative of Unit Vectors

    Quick question (a little rusty on this): Why don't unit vectors in Cartesian Coordinates not change with time? For example, suppose \mathbf{r} (t) = x(t) \mathbf{x} + y(t) \mathbf{y} + z(t) \mathbf{z} How exactly do we know that the unit vectors don't change with time? Or in other words...
  24. vetgirl1990

    Why Use Unit Vectors in Calculations?

    I'm confused about what situations you should use unit vectors in... and it seems that when I approach the same problem using unit vectors vs. without unit vectors, I get different answers. Why? To illustrate my confusion, here's an example that I tried solving using unit vectors, and without...
  25. R

    Physics: Multiplying Unit vectors

    [Moderator note: Post moved from New Member Introductions forum, so no template] I am having trouble understanding how to multiply unit vectors. I know that: (please excuse the notation) i^×j^ = k^ j^×k^ = i^ k^×i^ = j^ The question I am stuck on is: What is (i^×j^)×k^? So far I have (i^×j^)...
  26. X

    Unit vectors for multiple particles? (Quantum Mechanics)

    It's been a little bit since I have studied multi-particle quantum mechanics and I am a little rusty on the notation. Let's say I have a wave function, that consists of the tensor product of two spaces, one for each particle moving, ##|\psi_1,\psi_2>##. Each of these particles is moving in a...
  27. PhotonSSBM

    Perpendicular Unit Vectors in the x-y Plane: Is My Solution Correct?

    Homework Statement From Kleppner and Kolenkow Chapter 1 (Just checking to see if I'm right) Given vector A=<3, 4, -4> a) Find a unit vector B that lies in the x-y plane and is perpendicular to A. b) Find a unit vector C that is perpendicular to both A and B. c)Show that A is perpendicular to...
  28. R

    Unit Vectors for Polarization and Wave Vector Directions

    Homework Statement I am having difficulty understanding the very first step of the following solved problem (I understand the rest of the solution). How did they obtain the expressions for ##\hat{n}## (the direction of polarization), and ##\hat{k}## (the unit vector pointing in the direction...
  29. C

    Find Unit Vectors for f(x,y) w/ D_uf=0

    Homework Statement For f(x,y)=x^2-xy+y^2 and the vector u=i+j. ii)Find two unit vectors such D_vf=0 Homework Equations N/A. The Attempt at a Solution Not sure if relevant but the previous questions were asking for the unit vector u - which I got \hat{u}=\frac{1}{\sqrt{2}}(i+j) for the maximum...
  30. C

    Cartesian to polar unit vectors + Linear Combination

    I've been trying to solve this question all day. If somebody could point me in the right direction I would really appreciate it! (ii) A particle’s motion is described by the following position vector r(t) = 4txˆ + (10t − t)ˆy Determine the polar coordinate unit vectors ˆr and ˆθ for r. [4]...
  31. I

    Choosing unit vectors for harmonic motion problems

    Consider a vertical pendulum affected by gravity (See the pdf file i included). Now i can choose two different opposite directions for my unit vectors which give me different equations. \downarrow : m\ddot x = mg-kx \uparrow : m\ddot x = kx-mg Which of course makes perfect sense, changing...
  32. C

    Deriving spherical unit vectors in terms of cartesian unit vectors

    I'm trying to find the azimuthal angle unit vector \vec{\phi} in the cartesian basis by taking the cross product of the radial and \vec{z} unit vectors. \vec{z} \times \vec{r} = <0, 0, 1> \times <sin(\theta)cos(\phi), sin(\theta)sin(\phi), cos(\theta)> = <-sin(\theta)sin(\phi)...
  33. C

    Cartesian unit vectors expressed by Cylindrical unit vectors

    please someone explain me the following expression for Cartesian unit vectors expressed by the cylindrical unit vectors: http://web.mit.edu/8.02t/www/materials/modules/ReviewB.pdf at page B-8 line B.2.4 i would like to know which steps led to it. thanks, Chen
  34. deedsy

    Deriving sin(a-b) trig identity using Cross Product of Unit Vectors

    Homework Statement A and B are two unit vectors in the x-y plane. A = <cos(a), sin(a)> B = <cos(b), sin(b)> I need to derive the trig identity: sin(a-b) = sin(a) cos(b) - sin(b) cos (a) I'm told to do it using the properties of the cross product A x B Homework Equations A x B =...
  35. Ackbach

    LaTeX Best Unit Vectors in LaTeX for TeX.SE Interaction

    During an interaction on TeX.SE, egreg there posted some truly awesome code for doing unit vectors in $\LaTeX$: \usepackage{newtxtext} \usepackage{newtxmath} \usepackage{amsmath} \usepackage{bm} \newcommand{\uveci}{{\bm{\hat{\textnormal{\bfseries\i}}}}}...
  36. M

    Example about tangential and normal unit vectors

    Here is a example 1.3 from analytical dynamics of Haim Baruh. a particle moves on a path on the xy plane defined by the curve y=3*x^2 , where x varies with the relation x= sin(a). find the radius of curvature of the path and unit vectors in the normal and tangential directions when a=pi/6...
  37. F

    Integration including unit vectors

    I have an integral of aΘ cos(Θ) dΘ a is the unit vector for Θ. I'm not sure what to do with it in the integration. I know the unit vector equals a/abs(a) but that would give a mess of an integral cause of the abs(a).
  38. nomadreid

    States are or aren't unit vectors?

    I am a little confused by an elementary point. Something must be wrong with the following: On one hand, a Hermitian operator (which is not necessarily unitary) takes one state to another state. Hence a state need not be represented as a unit vector; its norm can be greater (or less than)...
  39. JasonHathaway

    Unit vectors in different coordinates

    Hi everyone, I've some points I want to make sure of. 1- When converting a "POINT" from a coordinate system to another, I'll just use the derived equation to convert (e.g. (1,2,3) from cartestian to cylindrical: \rho=\sqrt{x^{2}+y^{2}}, \phi=tan^{-1}\frac{y}{x}, z=z 2- When converting an...
  40. A

    Cross products for unit vectors in other coordinate systems

    I am a bit confused often when I have to compute cross products in other coordinate systems (non-Cartesian), I can't seem to find any tables for cross products such as "phi X rho." in spherical I think that these unit vectors are considered to be "perpendicular," so would phi X rho just be "+/-...
  41. JasonHathaway

    Understanding Unit Vectors: A Step-by-Step Guide

    Hi everyone, Just want to know how does the the unit vector become in that form: \vec{n}=\frac{2x\vec{i}+2y\vec{j}}{\sqrt{(2x)^{2}+(2y)^{2}}}=\frac{x \vec{i}+y \vec{j}}{4}
  42. E

    Derivation of Phi-Hat wrt Phi in Spherical Unit Vectors

    Homework Statement I just want to know how to get from this: ∂ø^/∂ø = -x^cosø - y^sinø to this: = -(r^sinθ+θ^cosθ) Homework Equations All the equations found here in the Spherical Coordinates section: http://en.wikipedia.org/wiki/Unit_vector The Attempt at a Solution I've...
  43. T

    What Is the Dot Product of Two Parallel Unit Vectors?

    Homework Statement The dot product for two.parralel pointing.unit.vectors is ? A. 1 B. 0 C. -1 D. Undefined [b]2. Relevant equation The Attempt at a Solutionsince they are unit vectors they have a magnitude of 1,this implies that the dot product is 1,since the angle between...
  44. jdawg

    What's the Difference Between Ax and i-Hat in Vector Notation?

    Homework Statement So this isn't really a specific homework question, it's more of a general one. What is the difference between ax and i(hat)? I thought they were the same thing. Can someone please explain the difference? Homework Equations The Attempt at a Solution
  45. Ascendant78

    Understanding Vectors vs Unit Vectors: Differences and Uses in Physics

    I am a bit confused about what the difference is between the two? To give some specific context where it has thrown me off, say if I were to define a charge with a vector r and compared that to a unit vector r hat, what exactly is the difference between what each of those tells me? I have...
  46. skate_nerd

    MHB Showing relationship between cartesian and spherical unit vectors

    I am asked to show that when \(\hat{e_r}\), \(\hat{e_\theta}\), and \(\hat{e_\phi}\) are unit vectors in spherical coordinates, that the cartesian unit vectors $$\hat{i} = \sin{\phi}\cos{\theta}\hat{e_r} + \cos{\phi}\cos{\theta}\hat{e_\phi} - \sin{\theta}\hat{e_\theta}$$ $$\hat{j} =...
  47. Raerin

    MHB If a and b are unit vectors....

    If a and b are unit vectors and |a + b| = sqrt(2). What is the value (dot product) of (2a-b).(a+3b)? Is the answer -1 by any chance? If not... I know how to find the dot product and find the magnitude and add vectors, etc. but I have never came across this a question before. I am very unclear...
  48. B

    Calculation of work involving unit vectors

    Homework Statement A loaded grocery cart is rolling across a parking lot in a strong wind. You apply a constant force f = (30N)i - (40N)j to the cart as it undergoes a displacement s = (-9.0m)i - (3.0m)j How much work does the force you apply do on the grocery cart? Homework Equations...