What is Unit vector: Definition and 167 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.
First, I use the unit vector of each plane, and I compute their crossproduct to obtain a vector parallel to the line of interception.
Then, I algebraically use x=0 to obtain the coordinates of the point in the line of interception. However, not having a y coordinate in plane one is confusing me.
Not sure how to show that because ##\vec{v} = v\hat{v} = 3e\hat{e}##, but since ##\vec{e}## is a unit vector we know ##e = 1## so our equation now becomes ##\hat{v} = \frac{3\hat{e}}{v}##. So, we're left to the task of showing that ##v = 3## in order to conclude that ##\hat{v} =...
I looked at this question and i wanted to ask if we could also use; ##C## =## c_2 ##(##\dfrac {3}{2}i## +## j  3k)## ... cheers
This problem can also be solved by using the approach of cross product ##A×B##...
So when evaluation the cross product of the velocity of the charge and the unit vectors associated with the point I am getting
v x r = j x [ i + j].
Well j x j is 0.
j x i = k, but yet the answer is positive. Why is this?
Considering an stopped object in a horizontal plane, the frictional force between the object and the plane would be the product of the friction coefficient (static or kinetic if there was movement between the surfaces) by normal. Since the normal in this case would be given by N (vector) =  mg...
I'm having difficulties solving this. For finding a unit vector that is orthogonal to two unit vectors I understand we use the cross product and such. However, I am confused about how to approach this problem as it has a third vector.
We can let x = (x1, x2, x3, x4) be a vector orthogonal to u...
A unit vector, ##\frac{\vec{v}}{\vec{v}}##, has dimensions of ##\frac{L}{L} = 1##, i.e. it is dimensionless. It has magnitude of 1, no units.
For a physical coordinate system, the coordinate functions ##x^i## have some units of length, e.g. ##\vec{x} = (3\text{cm})\hat{x}_1 +...
So I'm trying to figure out the integral of phi hat with respect to phi in cylindrical coordinates. My assumption was that the unit vector would just pass through my integral... is that correct? (I reached this point in life without ever thinking about how vectors go through integrals, and...
Summary: Meaning of each member being a unit vector, and how the products of each tensor can be averaged.
Hello!
I am struggling with understanding the meaning of "each member is a unit vector":
I can see that N would represent the number of samples, and the pointy bracket represents an...
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
The system considers a torus that has a wire wrapped around it, through which a current flows. In this way, a field originates in the phi direction.
The direction of current is "theta" in the spherical coordinate system but in toroidal system, in several book shows that the electrical current...
I am simulating random angles from 0 to 2π with a uniform distribution. However, if I take the differences between random angles, I get a nonuniform (monotonically decreasing) distribution of angles.
In math speek:
Ai = uniform(0,2π)
dA = Ai  Aj
dA is not uniform.
Here is a rough image of...
Homework Statement
A 0.54 kg block of ice is sliding by you on a very slippery floor at 2.1 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 0.0034 seconds. The block eventually slides at an angle of 21 degrees from its original...
Homework Statement
Take ∂2E/∂t2 E(r,t)=E0cos((k(u^·r−ct)+φ) in which u^ is a unit vector.
Homework Equations
d/dx(cosx)=sinx
The Attempt at a Solution
I had calc 3 four years ago and can't for the life of me remember how to differentiate the unit vector. I came up with...
When doing integration such as \int_{0}^{2\pi} \hat{\rho} d\phi which would give us 2\pi \hat{\rho} , must we decompose \hat{ρ} into sin(\phi) \hat{i} + cos(\phi) \hat{j} , then \int_{0}^{2\pi} (sin(\phi) \hat{i} + cos(\phi)\hat{j}) d\phi , which would give us 0 instead?
Thanks
Homework Statement
Givens:
An object with a mass of 2kg has a momentum of p=<1,2,3>. The first two questions asked for the magnitude of the momentum and the corresponding unit vector, which i found to be 3.74 and <0.267,0.535,0.802> respectively. The next question asks for the speed of the...
Homework Statement
Hi
Given the linear velocity formula: v* v^ = r*ω(sinθi^ + cosθj^)
i^, j^, v^  unit vectors
I'm to prove that v^ has direction, turn and magnitude
Magnitude:
v^ = sqrt((sinθ)^2 + (cosθ)^2) = 1 (as is also stated in unit vector's definition)
Direction and turn...
I need to solve:
\dot{\mathbf{r}}=kv\hat{r}  \dot{\mathbf{r}_s}
However, I do not know how to deal with the fact that there is a unit vector. How can this be done? \dot{\mathbf{r}_s} is a constant vector.
Homework Statement
Find the unit vector perpendicular to the level curve of f(x,y) = x2y10xy9y2 at (2,1)
Homework Equations
Gradient
The Attempt at a Solution
I'm not sure what it's asking. Wouldn't this just be the gradient of f(x,y) evaluated at (2,1) then normalized? or am I missing...
Homework Statement
Given ## d \vec r = dr \hat r + r d \theta \hat {\theta} + r \sin \theta d \phi \hat {\phi}.## Find ## d \hat r , d \hat {\theta} , d \hat {\phi}. ##
Homework Equations
I know that ## d \hat {e_j} = \omega^i_j \hat {e_i} ## and that ## \omega_{ij}= \omega_{ji} ## and ## 0 =...
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...
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...
Homework Statement
How does the unit vector have no units
I know that the unit vector has a length of 1 and zero units. A representation would be Ihat jhat khat(depending on the coordinate system). But the unit vector is the vector/magnitude. If the unit vector is its vector/magnitude then...
Homework Statement
I'm fully convinced that the zero values make sense, yet they are wrong, can somebody please explain why is that the case
Homework Equations
N/A
The Attempt at a Solution
Attempt in the image above
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?
So I'm reading the Schaum's outlines while trying to prepare for a big test I have in September. And I'm trying to understand something here that maybe someone can offer some clarification and guidance.
So, using Coulomb's Law, we can find the electric field as follows:
\begin{equation}
dE...
Hi everybody! I'm currently learning special relativity, and I'm going through the chapter of tangent, normal and binormal vectors. In my teacher's script, the definition of the normal vector eN says:
\vec{e_N} = \frac{d}{ds} \vec{e_T} \cdot \frac{1}{\mid \frac{d}{ds} \cdot \vec{e_T} \mid} =...
Homework Statement
Find two unit vectors in 2space that make an angle of 45(deg) with 7i + 6j.
Homework Equations
Unit Vector = u = (1/v)*v
Dot Product = u . v = u v cos(theta)
The Attempt at a Solution
I've been thinking about a way to do this. I originally thought that...
Homework Statement
I am supposed to find the tangent and normal unit vector to r(t)=<2sint,5t,cost>.Homework Equations
.
The Attempt at a Solution
r1(t)=<2cost,5,sint> which is a tangent vector to the curve, and then to make it a unit vector I would multiply by 1/(sqrt(4cos^2t+25+sin^2t)...
[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^)...
Homework Statement
Hi!
I have the 3x3 matrix for L below, which I calculated. But now I need to figure out how the equation below actually means! Is it just the inverse of L (L^1)? I cannot proceed if I don't know this step.
Homework Equations
See image
The Attempt at a Solution
I put in...
1. The problem statement, all variables and given/known dat
A car travels 20 mi at 60 degrees north of west, then 35 mi at 45 degrees north of east.
Express each displacement vector in unit vector notation. Take the +xaxis due east and the +yaxis due north. Use the component method to obtain...
Homework Statement
if a =3i4j
find a unit vector perpendicular a
Homework Equations
Vector
The Attempt at a Solution
<a> = <3 , 4>
<n> = <x, y> : x² + y² = 1
<a>•<n> = 3x4y = 0
y = (3/4)x
x² + (9/16)x² = 1
25x² = 16
x = 4/5, 4/5
y = 3/5, 3/5
There are two unit vectors normal...
Sorry if this was addressed in another thread, but I couldn't find a discussion of it in a preliminary search. If it is discussed elsewhere, I'll appreciate being directed to it.
Okay, well here's my question. If I take the divergence of the unit radial vector field, I get the result:
\vec...
http://tutorial.math.lamar.edu/Classes/CalcIII/CurlDivergence.aspx
In the above link, in the 2nd to last blue box on the page (where it tells how to solve a line integral with respect to a vector field using the curl), it says that the line integral of vector field F with respect to r (with an...
Hello every one i have question what is the unit vector of time component like c^2 t^2 x^2y^2z^2 = 0
OR
if we need to calculate the d'alembert cross some 4 vector have 3 space component x y z and unit vector i j k
and ct component what is the unit vector in Ct direction ?
Hi, I was wondering if anyone could help with a vector question that I have.
If I have a unit vector defined in cartesian coordinates as p= (0,1,0) how would I go about converting this vector to a cylindrical geometry.
I understand that I will probably need to use p_r=sqrt(px^2+py^2) and...
Hello! I need help with this problem
how do you find a unit vector along the direction $\vec{A}=2\hat{a_{\rho}}z\hat{a_{z}}$(cylindrical)?
do I have to convert it to Cartesian or there is a direct method? please help! Thanks!
I know that for two points, the difference between them is a line segment
lets say these two points are 'a' and 'b' respectively, so ba = "new vector represent the line"
In my textbook ba=d*t  where 'd' is a vector along the directon of 'ba' and t is a parameter.
does 'd' actually...
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 unitvector notation.
Homework Equations
x = rcos
y = rsin
The Attempt at...
Homework Statement
Suppose ##F(x, y, z, u, v) = xy^2 + yz^2 + zu^2 + uv^2 + vx^2 ## Standing at the point ##(1, 1, 1, 1, 1) ## imagine moving in a direction ##\vec w ## where ##\vec w ## is a unit vector. Find the components of a vector ##\vec u ## such that ## D_\vec u F = 0 ##
Remember...
Find in rectangular coordinates a unit vector which is: A. in the direction of E at P(2,3,4)if $\overline{E}=(x^2+y^2+z^2)\left(\frac{xa_{x}}{\sqrt{y^2+z^3}}+\frac{ya_{y}}{\sqrt{x^2+z^2}}+\frac{za_{z}}{\sqrt{x^2+y^2}}\right)$; B. Perpendicular to the plane passing through M(1,5,5), N(2,4,0)...
Homework Statement
An em wave in free space has an electric field vector E = f(tz/c0)x where x is a unit vector in the x direction and f(t)= exp(t2/τ2)exp(j2πv0t). Describe the physical nature of this wave and determine an expression of the magnetic field vector.
Homework Equations...
The problem:
The velocity ~v (vector notation, don't know how to type) is given by:
~v = (6.0t  4.0t^2)ˆi + 8.0ˆj + (3.0t) with ~v in meters per second and positive t seconds. ˆi, ˆj, ˆk have their usual meanings (unit vector notation).
(a) What is the acceleration ~a of the particle when...
Homework Statement
Given the ellipse
##0.084x^2 − 0.079xy + 0.107y^2 = 1 ##
Find the semimajor and semiminor axes of this ellipse, and a unit vector in the
direction of each axis.
I have calculated the semimajor and minor axes, I am just stuck on the final part.
Homework Equations
this...
Hello
I'm getting confused when I want to use magnetic boundary equation
could you tell me how we define the unit vector(an) in this equation?
for example you assume that we have two different region (A in red and B in yellow) which vector (1,2,3,4) is right for equation and which is right...
Homework Statement
electric field polarization for a given antenna is expressed as:
E_i = (a_x + a_y) E (r, theta, phi)
The unit vector along the antenna polarization is found as u_a = 1/Sqrt(2) (a_x + a_y)2. Question
Where/how is the 1/Sqrt(2) found to resolve those other 2 vectors into a...
Homework Statement
So the full problem reads: A vector F has the same magnitude and direction at all points in space. Choose the zaxis parallel to F. Then , in Cartesian coordinates, \vec{F}=F\hat{z}, where \hat{z} is the unit vector in the z direction. Express \vec{F} in spherical...