01.3차원 물체와 벡터 표현 - gobuksun/mywiki GitHub Wiki

1. VPython(Glowscript)을 이용한 3차원 물체 표현


좌표계



HelloPhysics

GlowScript 2.9 VPython


#Creating Ball
myBall = sphere(color=color.red, radius=2)

#Creating Box
myBox = box(pos = vec(5,0,0), color = color.blue, size = vec(0.5,4,1))

myBall.color = color.green
myBox.pos.x = 10




2. VPython(Glowscript)을 이용한 벡터 연산

벡터(Vector)

  • A vector has magnitude (size)크기 and direction방향

     
    The length of the line shows its magnitude and the arrowhead points in the direction.


    • We can add two vectors by joining them head-to-tail
    We can also subtract one vector from another
    Notation
    The vector a is broken up into the two vectors ax and ay


    Example: add the vectors a = (8, 13) and b = (26, 7)
    c = a + b
    c = (8, 13) + (26, 7) = (8+26, 13+7) = (34, 20)
    Example: subtract k = (4, 5) from v = (12, 2)
    a = v + −k
    a = (12, 2) + −(4, 5) = (12, 2) + (−4, −5) = (12−4, 2−5) = (8, −3)

  • Magnitude of a Vector
The magnitude of a vector is shown by two vertical bars on either side of the vector:
|a|

OR it can be written with double vertical bars (so as not to confuse it with absolute value):
||a||

We use Pythagoras' theorem to calculate it:
|a| = √( x2 + y2 )

Example: what is the magnitude of the vector b = (6, 8) ?
|b| = √( 62 + 82) = √( 36+64) = √100 = 10

A vector with magnitude 1 is called a Unit Vector.

  • Unit Vector
A Unit Vector has a magnitude of 1:

The symbol is usually a lowercase
letter with a "hat", such as:

(Pronounced "a-hat")

Scaling

A vector can be "scaled" off the unit vector. Here vector a is shown to be 2.5 times a unit vector. Notice they still point in the same direction:

 

myVector
GlowScript 2.9 VPython

#Vectors
r= vector(3,4,5)
r_arrow = arrow(pos = vector(0,0,0), axis=r, shafrwidth=0.2)

#Axes (사물의) (중심) 축, 축선
x_axis = arrow(axis=vector(10,0,0), color=color.red, shaftwidth=0.1)
y_axis = arrow(axis=vector(0,10,0), color=color.green, shaftwidth=0.1)
z_axis = arrow(axis=vector(0,0,10), color=color.blue, shaftwidth=0.1)

r_mag = mag(r)    // 벡터의 크기 : 스칼라값
r_hat = norm(r)   // 벡터의 방향 : 단위벡터
r_hat_arrow = arrow(axis=r_hat, color=color.cyan, shaftwidth=0.2)

print("r:", r)
print("r_mag:", r_mag)
print("r_hat:", r_hat)


벡터 연산

GlowScript 3.0 VPython

a = vector(3,4,0)
b = vector(5,1,0)

# 벡터더하기
c = a + b

# 벡터빼기
d = a - b

# 벡터의내적
e = dot(a,b)

deg = acos(e/mag(a)/mag(b)) ## rad
deg = degrees(deg) ## deg
a_arrow = arrow(pos=vector(0,0,0), axis=a, shaftwidth=0.2,color=color.blue)
b_arrow = arrow(pos=vector(0,0,0), axis=b, shaftwidth=0.2,color=color.green)
#c_arrow = arrow(pos=vector(0,0,0), axis=c, shaftwidth=0.2,color=color.red)
d_arrow = arrow(pos=b, axis=d, shaftwidth=0.2,color=color.red)

print("dot(a,b) =", e)
print("degree(a,b) =", deg)
print("a+b=", c)
print("a-b=", d)

벡터의 내적 : 벡터로 이루어진 물리량을 분해할때 많이 쓰임, 힘을 분해한다든지, 속도를 분해한다든지...

Multiplying a Vector by a Vector (Dot Product and Cross Product)

dot product magnitude and angle

How do we multiply two vectors together? There is more than one way!

(Read those pages for more details.)


myVector-cross

GlowScript 2.9 VPython

#Vectors
a = vector(1,2,3)
b = vector(-1,2,0)
c = cross(a,b)

a_arr = arrow(axis=a, shaftwidth=0.1, color=color.red)
b_arr = arrow(axis=b, shaftwidth=0.1, color=color.blue)
c_arr = arrow(axis=c, shaftwidth=0.1)

d = cross(b,a)
d_arr = arrow(axis=d, shaftwidth=0.1, color=color.yellow)


myVector-dot

GlowScript 2.9 VPython

#Vectors
v = vector(3,4,0)
r = vector(1,0,0)
v_par = dot(v,r)*r
v_per = v - v_par
v_arr = arrow(axis=v, shaftwidth=0.2)
r_arr = arrow(axis=r, shaftwidth=-0.3, color=color.blue)
v_par_arr = arrow(axis=v_par, shaftwidth=0.2, color= color.green)
v_per_arr = arrow(axis=v_per, shaftwidth=0.2, color= color.yellow)

print("v=", v)
print("r=", r)
print("dot(v,r)=", dot(v,r))
print("v_par=", v_par)
print("v_per=", v_per)





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