Euclidean vector

Euclidean vector

A

Euclidean vector

is a geometric entity that has the of magnitude and

direction

. For example, a vector in

ℝ

2

can be represented by its

components

like this

(

3

,

4

)

or like this

[

3

4

]

. This particular vector can be represented geometrically like this:

Figure 1:

Geometrical representation of a Euclidean vector

In

ℝ

n

, a vector can be easily constructed as the

line segment

between points whose in each

coordinate

are the components of the vector. A vector constructed at the origin (The vector

(

3

,

4

)

drawn from the point

(

0

,

0

)

to

(

3

,

4

)

) is called a

position vector

. Note that a vector that is not a position vector is of position. It remains the same vector unless one changes its magnitude or direction.

Magnitude:

The magnitude of a vector is the from one end of the line segment to the other. The magnitude of the vector comes from the metric of the space it is embedded in. It’s magnitude is also referred to as the

vector norm

. In

Euclidean space

, this can be gotten using Pythagorean’s

theorem

in

ℝ

n

such that for a vector

v

→

∈

ℝ

n

, the length

|

v

→

|

is such that:

|

v

→

|

=

∑

i

=

0

n

x

n

2

This can also be found by the

dot product

a

→

⋅

a

→

.

Direction:

A vector basically

is

a direction. However you may want to know the angle it makes with another vector. In this case, one can use the dot product for the simplest computation, since if you have two vectors

a

→

,

b

→

,

a

→

⋅

b

→

=

|

a

→

|

⁢

|

b

→

|

⁢

cos

⁡

θ

Since

θ

is the angle between the vectors, you just solve for

θ

. Note that if you want to find the angle the vector makes with one of the axes, you can use

trigonometry

.

Figure 2:

Angle between a vector and the axis

In this case, the length of the vector is just the

hypotenuse

of a single

right triangle

, and the angle

θ

is just

arctan

⁡

y

x

, the arctangent of it’s components.

, resolving a vector into its components:

Say all we had was the magnitude and angle of the vector to the x-axis and we needed the components.(Maybe we need to do a

cross product

or

addition

). Look at figure a. again. The x component of the vector can be likened to a “plumb line” dropped from the arrowhead of the vector to the x-axis so that it is

perpendicular

to

the x-axis. Thus, to get the x-component, all we would have to do is multiply the magnitude of the vector by the cosine of

θ

. This is called resolving the vector into its components.

Now say we had another vector at an angle

θ

2

to our vector. We construct a line between the of our vector and intercept the other vector in the same way, perpendicularly. The length from the tail of the vector to our interception is called the

of our vector onto the other. Note that the equation remains the same. The is still

d

⁢

cos

⁡

θ

2

. This is valid in all higher

dimensions

as the

angle between two vectors

takes place on a plane.

Figure 3:

Projection

of one vector onto another

Miscellaneous :

⋅

Two vectors that are

parallel

to each other are called “

collinear

”, as they can be drawn onto the same line. Collinear vectors are

scalar multiples

of each other.

⋅

Two non-collinear vectors are

coplanar

.

⋅

There are two main of products between vectors. The dot product (

scalar product

), and the cross product (vector product). There is also a commonly used called the

triple scalar product

.

Title

Euclidean vector

Canonical

name

EuclideanVector

Date of creation

2013-03-22 11:58:47

Last modified on

2013-03-22 11:58:47

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

29

Author

mathwizard (128)

Entry type

Definition

Classification

msc 15A72

Synonym

Euclidean vector properties

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