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
v
^
{\displaystyle {\hat {\mathbf {v} }}}
(pronounced "vhat").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 nonzero vector u is the unit vector in the direction of u, i.e.,
u
^
=
u

u

{\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 threedimensional 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 righthand rule to resolve one of two possible directions.
"Firstly, I represented [Uθ ]on the twodimensional 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...
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
(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 =...
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...
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...
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...
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.
Homework Statement: Hallo. Can somebody explain to me what's the importanceuse 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
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...
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...
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...
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...
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 crossproduct of ##\hat{ρ}## and ##\hat{z}##...
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...
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 cosθersinθeθ, but I don't get ex correctly.
am i doing this wrong?
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 
VfVi = √(32+52+72)√(12+52+22) = 3.63...
Homework Statement
Homework Equations
The Attempt at a Solution
So I began by subtracting.
(205160)=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...
$\tiny{s6.12.3.35}\\$
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?
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)...
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...
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...
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...
[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^)...
It's been a little bit since I have studied multiparticle 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...
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 xy 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...
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...
Homework Statement
For f(x,y)=x^2xy+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...
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]...
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 = mgkx
\uparrow : m\ddot x = kxmg
Which of course makes perfect sense, changing...
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)...
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 B8 line B.2.4
i would like to know which steps led to it.
thanks,
Chen
Homework Statement
A and B are two unit vectors in the xy plane.
A = <cos(a), sin(a)>
B = <cos(b), sin(b)>
I need to derive the trig identity:
sin(ab) = 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 =...
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}}}}}...
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...
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).
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)...
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...
I am a bit confused often when I have to compute cross products in other coordinate systems (nonCartesian), 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 "+/...
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}
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...
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...
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
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...
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} =...
If a and b are unit vectors and a + b = sqrt(2). What is the value (dot product) of (2ab).(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...
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...