Someone Knows what is unit vector and I'm a math idiot, so could you make a detailed example?

Question

Someone Knows what is unit vector and I'm a math idiot, so could you make a detailed example?

Many thanks~

Answer 1

Let me simplify it
Take a vector, for example, it is 3i + 4 j + 0k ( that is, 3 units to the right parallel to X-axis, 4 units up parallel toY -axis and no units into the paper on the Z a-xis )

The vector, you divide it by its own magnitude, that is, the length of the vector itself, its total length in this case is root of ( 3sq + 4sq ), by pythagoras theorem.
Which is 5 units.

So the unit vector would be (1/5).( 3 i + 4 j + 0 k )

And the thing about unit vectors is that their magnitude is always 1 unit, and they have the same direction as the vector.

Answer 2

A vector has both length (magnitude) and direction for example 30 km South is a vector though 30 km on its own is a scalar because there is no direction. Think of a line drawn on a page with an arrow indicating direction. If the lenth of the line is 1 unit it is a unit vector.

Answer 3

A unit vector in a normed vector space is a vector (often a spatial vector) whose length is 1. A unit vector is often written with a superscribed caret or “hat”, thus: {\hat{i}} (pronounced "i-hat").

In Euclidean space, the dot product of two unit vectors is simply the cosine of the angle between them. This follows from the formula for the dot product, since the lengths are both 1.

The normalized vector {\hat{u}} of a non-zero vector u is the unit vector codirectional with u, i.e.,

\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.

The elements of a basis are usually chosen to be unit vectors. Every vector in the space may be written as a linear combination of unit vectors, with the components of each being given by direction cosines. The most commonly encountered bases are Cartesian, polar, and spherical coordinates. Each uses different unit vectors according to the symmetry of the coordinate system. Since these systems are encountered in so many different contexts, it is not uncommon to encounter different naming conventions than those used here. Usually, a little context should enable the astute reader to substitute the names being used for those given here.

Answer 4

A unit vector is any vector in a normed vector space with a length of 1. Graphed, it would extend from the origin to some point on the unit circle/sphere/higher dimensional analouge.

Answer 5

A vector group reference when applied to say a the magnetic field of a transformer, identifies the phase difference as an angle for each of the three phases.
It describes the lead or lag of this angle from unity.
To make a detailed example is difficult unless you know the subject.
Anyway at 83 I would be pushed to do it now.

Answer 6

A unit vector of a given vector is a vector of unit magnitude and has the same direction as that of the given vector.

Unit Vector = Vector/Modulus of vector

Answer 7

A unit vector is a vector with a length of 1(unit). this vector can be calculated by dividing a vector to it's length. for example in a 3d space using Cartesian coordinate system, you have three perpendicular unit vectors: i= [1,0,0] ,,j= [0,1,0] , k= [0,0,1]. Other coordinate systems, such as polar coordinates or spherical coordinates use different unit vectors;

Answer 8

A unit vector is a vector of length one. Usually denoted i, j, k for unit vectors in the direction of x, y, z respectively. You can then describe any vector by a combination of i, j, k. A vector pointing from (0,0,0) to (1,2,3) is labled 1i + 2j + 3k.

i, j, k are usually bold or underlined to show they are vectors.

Answer 9

Unit vector = vector/magnitude of the vector
For example
Let two points on a plane are A (5,4) and B(7,5)
Vector AB=(7 - 5)i +( 5 - 4)j
=2i+j...............................................ii
the modulus or magnitude = sq-rt {(2)^2 +(1)^2}
=sq-rt(5)..................ii
Unit vector = (2i+j)/sq-rt 5