# 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 "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.,

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 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. ### Complete the parametric equation for the line where the planes cross

First, I use the unit vector of each plane, and I compute their cross-product 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.
2. ### Vector problem: Questions about a unit vector

Not sure how to show that because ##\vec{v} = |v|\hat{v} = 3|e|\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} =...
3. ### Determine a unit vector perpendicular to the given planes

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##...
4. ### Help with unit vector for a magnetic field

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?
5. ### Vectorial issue of friction

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...
6. ### Finding Orthogonal Unit Vector to 3 Vectors

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...
7. E

### B The length of a unit vector?

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 +...
8. ### What's the integral of a unit vector?

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...
9. ### I Meaning of each member being a unit vector

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...
10. ### 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

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12. ### Doubt about a unit vector in toroidal coordinates

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...
13. ### I Random Unit Vector Angle Difference

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 non-uniform (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...
14. ### Finding z component of a unit vector

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...
15. ### B Confusion about the radius unit vector in spherical coordinates

If the radius unit vector is giving us some direction in spherical coordinates, why do we need the angle vectors or vice versa?
16. ### Derivative of Cosine with unit vector

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...
17. ### I Integrating unit vector ρ

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
18. ### What is the speed of an object with a momentum of <-1,-2,3>?

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...
19. ### Prove v^ has all of a vector's quantities

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...
20. ### B Solving a differential equation with a unit vector in it

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.
21. ### Unit vector perpendicular to the level curve at point

Homework Statement Find the unit vector perpendicular to the level curve of f(x,y) = x2y-10xy-9y2 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...
22. ### Derivative of unit vector in spherical coords.

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 =...
23. ### 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...
24. ### 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...
25. ### Unit vector questions

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 I-hat j-hat k-hat(depending on the coordinate system). But the unit vector is the vector/magnitude. If the unit vector is its vector/magnitude then...
26. ### Body diagonals -- unit vector notation

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
27. ### 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?
28. ### I How to write the unit vector for the spherical coordinates

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: dE...
29. ### I Normal Vector & Acceleration: An Explanation for Julien

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} =...
30. ### Find a unit vector for a particular vector

Homework Statement Find two unit vectors in 2-space 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...
31. M

### Normal Unit Vector to Curve

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)...
32. ### 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^)...
33. ### 3x3 matrix inverse unit vector

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...
34. ### Find Resultant Displacement Vector in Unit Vector Notation

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 +x-axis due east and the +y-axis due north. Use the component method to obtain...
35. ### Unit Vector Perp. to a: Solving Problem

Homework Statement if a =3i-4j 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> = 3x-4y = 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...
36. ### Divergence of radial unit vector field

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...
37. ### Standard unit vector in the positive z direction?

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...
38. ### Time Unit Vector: Calculate d'Alembert with 4 Vectors

Hello every one i have question what is the unit vector of time component like c^2 t^2 -x^2-y^2-z^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 ?
39. ### Converting a unit vector from cartesian to cylindrical

Hi, I was wondering if anyone could help with a vector question that I have. If I have a unit vector defined in cartesian co-ordinates 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...
40. ### MHB Finding a Unit Vector Along $\vec{A}$ - Please Help!

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!
41. ### Unit vector of a line in straight line equation

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 b-a = "new vector represent the line" In my textbook b-a=d*t -- where 'd' is a vector along the directon of 'b-a' and t is a parameter. does 'd' actually...
42. ### Vector and component vectors

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...
43. ### Finding the unit vector

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...
44. ### MHB Unit vector perpendicular to a plane.

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)...
45. ### Magnetic field from electric field given a function of time

Homework Statement An em wave in free space has an electric field vector E = f(t-z/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...
46. ### Velocity function, unit vector notation, acceleration, speed

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...
47. ### Finding the unit vector for an ellipse

Homework Statement Given the ellipse ##0.084x^2 − 0.079xy + 0.107y^2 = 1 ## Find the semi-major and semi-minor axes of this ellipse, and a unit vector in the direction of each axis. I have calculated the semi-major and minor axes, I am just stuck on the final part. Homework Equations this...
48. ### Unit vector in Magnetic boundary condition

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...
49. ### Resolving vectors into a unit vector

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...
50. ### A vector F=Fz, where z is unit vector, expressed in sphereical coord.

Homework Statement So the full problem reads: A vector F has the same magnitude and direction at all points in space. Choose the z-axis 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...