Vector

what can u tell about vectors?

a vector is an element of a vector space

du kannst mich mal am vector lecken!

sascha?

hoert sofort auf zu trollen -.-

desu

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a vector is imaginary

>>2

As this guy said. Also, we can then apply properties and basis to the vector space. Most people are generally familiar with vectors in normal euclidean vector space of the basis (1,0) and (0,1) which they then apply properties (for applications) like centimeters for distance, or centimeters per second for velocity, and many others (eg, a car is has a acceleration vector pointed in the direction it's tires are facing which is measured in the units meters per second per second (m/s^2).

>>9
THE TIRES MAY NOT ALL BE FACING THE SAME DIRECTION, USELESS DEFINITION OF CARVECTORS

A vector is a quantity with magnitude and direction.

>>11
tell us about the magnitude and direction of the C^inf vector \x -> sin(x) then.

>>10

I was making an analogy, asshole, the physics are actually a lot more complicated and subtle with air resistance vectors coming in the opposing direction, tangent vectors for the centripetal forces, gravity vectors working with friction to manifest as a psuedo counter-force vector, and so on and so forth (still generalized).

>>12
magnitude is |sin(x)|, direction is sin(x).

lucy has vectors

>>11

gtpho physphag

>>11

Fail fail fail.

The only things you need from a vector are the ability to add two vectors of the same type (in the same vector space) together, or multiply by a scalar constant, and for the result to be another vector. That is ALL YOU NEED

Read the books suggested by

http://dis.4chan.org/read/sci/1177621225/1-

A vector is an agent used to insert genetic information into a living cell.

A vector is an agent used to insert cum into your mom's bodily orifices

a vector is an element, commonly an ordered n-tuple, in a set, commonly an n-dimensional space, satisfying the properties of a vector space.
V is a vector space over the field F if:
it has two operations defined on it, vector addition, and scalar multiplication.
it satisfies the following properties:
associative addition of vectors
commutative addition of vectors
there is a "zero" vector satisfying additive identity
every vector has an additive inverse
distributivity of scalar multiplication, c(u+v) = cu + cv
distributivity of scalar multiplication, (c+d)u = cu + du
a(bv) = (ab)v for a,b in F, v in V
scalar multiplication has an identity element

V is necessarily closed under vector addition and scalar multiplication

also
>>11
>>13
fail.

A vector is a number that generally has more than one dimension.

>>21

Thread over.

Vectars??

The dot product is the product of the cosine of the angle between two vectors and their length/magnitude.
It can be written it as:

A . B = |A||B|cos theta

• Its clear that A.B = B.A (commutativity)
• also (kA).B = k(A.B), for any scalar k
• Its should also be noted that A.(B+C) = A.B + A.C (distributive)

Now define 3 perpendicular vectors i, j and k, we can now state


i.i = j.j = k.k = 1
i.j = j.k = k.i = 0

if we have our vectors A and B in component form:

A = i*x1 + j*y1 + k*z1
B = i*x2 + j*y2 + k*z2

We can then find a compenent form expression for the dot product.


A.B = (i*x1 + j*y1 + k*z1).(i*x2 + j*y2 + k*z2)

    = (i*x1 + j*y1 + k*z1).(i*x2)
    + (i*x1 + j*y1 + k*z1).(j*y2)
    + (i*x1 + j*y1 + k*z1).(k*z2)

    = (i*x1).(i*x2) + (j*y1).(i*x2) + (k*z1).(i*x2)
    + (i*x1).(j*y2) + (j*y1).(j*y2) + (k*z1).(j*y2)
    + (i*x1).(k*z2) + (j*y1).(k*z2) + (k*z1).(k*z2)

    = (i.i)*x1*x2 + (j.i)*y1*x2 + (k.i)*z1*x2
    + (i.j)*x1*y2 + (j.j)*y1*y2 + (k.j)*z1*y2
    + (i.k)*x1*z2 + (j.k)*y1*z2 + (k.k)*z1*z2

    = (1)*x1*x2 + (0)*y1*x2 + (0)*z1*x2
    + (0)*x1*y2 + (1)*y1*y2 + (0)*z1*y2
    + (0)*x1*z2 + (0)*y1*z2 + (1)*z1*z2

    = x1*x2 + y1*y2 + z1*z2

Now the cross product is defined as the product of the sine of the angle between two vectors, their magnitude and a unit vector perpendicular to those vectors.


AxB = |A||B|sin(theta)n
    = (i*x1 + j*y1 + k*z1)×(i*x2 + j*y2 + k*z2)

we can make the following observations about the cross of the vectors i, j, k


x| i  j  k
----------
i| 0  k -j
j|-k  0  i
k| j -i  0

it should be noted this table can be made in two ways, left or right handed. So there are really two definitions of cross product but its conventional to use this one.

• The cross product is distributive like the dot product.

again in component form:

A×B = (i*×1 + j*y1 + k*z1)×(i*×2 + j*y2 + k*z2)

    = (i*×1 + j*y1 + k*z1)×(i*×2)
    + (i*×1 + j*y1 + k*z1)×(j*y2)
    + (i*×1 + j*y1 + k*z1)×(k*z2)

    = (i*×1 + j*y1 + k*z1)×(i*×2)
    + (i*×1 + j*y1 + k*z1)×(j*y2)
    + (i*×1 + j*y1 + k*z1)×(k*z2)

    = (i*×1)×(i*×2) + (j*y1)×(i*×2) + (k*z1)×(i*×2)
    + (i*×1)×(j*y2) + (j*y1)×(j*y2) + (k*z1)×(j*y2)
    + (i*×1)×(j*z2) + (j*y1)×(k*z2) + (k*z1)×(k*z2)

    =     0*×1*×2 +    -k*y1*×2 +     j*z1*×2
    +     k*×1*y2 +     0*y1*y2 +    -i*z1*y2
    +    -j*×1*z2 +     i*y1*z2 +     0*z1*z2

    = i*(y1*z2-z1*y2) + j*(×1*y2-y1*×2) + k*(×1*y2-y1*×2)

now if you have ever done any matrix math, you may notice that this is equal to the determinant of the following 3x3 matrix

/x1 y1 z1\
|x2 y2 z2|
\ i  j  k/

>>24
You're thinking too physically.  What you state are properties of a specific vector space (namely R^3).  >>21 pretty much defined what a vector space is, and vectors are (obviously) elements of a vector space.  Some of the properties continue to hold in other vector spaces, while some do not.  Other than that, good attempt nevertheless.

>>25
fuck off.

>>27
fuck you

1 by n or n by 1 matrix

>>28
wrong

ATTN DUMBSHITS:

Vector is a meal replacement.

I feel the need, the need for weed!

Marijuana MUST be legalized.