Mindmap

VECTORS

C D MATHS Englisch

Inhalt der Mindmap

VECTORS

1.magnitude 2.direction

PLANE VECTOR~(自学,待删减)
- 向量的运算
  - 实数与向量的积: 1.|λa|=|λ||a|, 2.λ>0,方向一致. λ<0方向相反. λ=0 λa=0.
  - 向量的加减运算: a+b=(x1+x2,y1+y2), a-b=(x1-x2,y1-y2)
  - 有向线段: AB=(x2-x1,y2-y1)
  - 平面向量数量积 (scalar/ dot product): ab=|a||b|cosθ=x1x2+y1y2
    - Application
      - b在a上的投影: |b|cosθ=ab/|a|
        ab的几何意义: 数量积ab等于|a|与b在a上的投影的积
      - a⊥b<-->ab=0, x1x2+y1y2=0
      - cosθ=ab/(|a||b|), θi=arcos[ab/(|a||b|)] ??
  - 向量的模: a^2=|a|^2=x^2+y^2<-->|a|=√(x^2+y^2)
  - 两点间的距离: |AB|=√[(x2-x1)^2+(y2-y1)^2]
  - 平面向量的矢量积 (vector/ cross product): a×b=|a||b|(sinθ)п, b×a=|b||a|sinθ(-п)=-|a||b|(sinθ)п=-a×b; a×b=| in determinant form |
    - eg. j×k=1×1×(sin90)×i=i, a×b=0 then either a=0 or b=0 or a and b are parallel
    - Application.
      - sinθ=|a×b|/(|a||b|), θi=arcsin[|a×b|/(|a||b|)]
      - interpret area of triangle, parallelogram: At=1/2|a×b|=1/2|AB×AC|=1/2|(b-a)×(c-a)|=1/2|(a×b)+(b×c)+(c×a)|, Ap=|AB×AD|=|(b-a)×(d-a)|=1/2|(a×d)+(b×d)+(d×a)|
  - θ: angle between two vectors
  - 平面向量的混合积 (triple scalar product): λ(a×b)=| in determinant form |
    - Application
      - interpret volume of parallelepiped, tetrahedron ??
- 向量的运算律
  - Commutative: a+b=b+a, λ(μa)=(λμ)a, ab=ba, a×b≠b×a
  - Associative: a+b+c=(a+b)+c, a-b-c=a-(b+c), (λa)b=λ(ab)=a(λb), a(bc)≠(ab)c
  - Distributive: (λ+μ)a=λa+μa, λ(a+b)=λa+λb, (a+b)c=ab+ac
- 向量平行与垂直的充要条件
- 向量的概念
  - 向量的模才能比较大小
  - 零向量 0: 长度为0,方向任意
  - 单位向量: |v|=1
  - 平行向量(共线向量): a//b, 方向无关
  - 相等向量: 长度相等且方向相同的向量. 经过平移总可以重合: (x1,y1)=(x2,y2)<-->x1=x2, y1=y2
VECTOR GEOMETRY
- triple scalar product
  - a×(bc): bc--> scalar , vector 不能投影在scalar 上??
  - relevant properties: 1.cyclic interchange(a,b,c) : a(b×c)=b(c×a)=c(a×b), 2.a(a×b)=0 while θ=90--> cosθ=0, 3.also can expressed as {a,b,c} while order matters
  - difinition: a(b×c), geometrically (the magnitude) is the volume of parallelepiped between them. while aп= acosф
    - volume of tetrahedron= 1/6 volume of parallelpiped. since Vt=1/3 Area of base × h= 1/6 {a,b,c}= 1/6 {AB, AC, AD} (for tetrahedron ABCD)
      - 在实际应用中, 需要找两点得到一个vector. 
  - component form
- scalar product
  - or dot product('dot' in the middle) or scalar product(only scalar remained. name source.) it scales the vector. changes the magnitude of vector but leaves its direction unchanged.
  - definition: ab=|a|(|b|cosθ), geometrically is the magnitude of one times the projection of the second onto the first.
  - distributive and commutative
  - revelant: 1. |ab|=|a||b|cosθ in magnitude
- vector product - beautiful as it only needs coordinates
  - or cross product(large cross) or vector product(have both m&d)
  - definition: a×b=|a||b|sinθп, п is the unite vector perpendicular to both a and b and in the direction specified by right-hand screw rule
    - (个人理解)  力与力臂(他们俩在一个平面上)造成第三个方向的作用(另一个平面上). e.g.螺丝和┌扳手: 顺时针旋转扳手时,造成的作用: 螺丝进去, 视为 + 方向作用螺丝出来, 造成 - 方向作用
  - relevant corollaries: 1. |a×b|=|a||b|sinθ in magnitude 2.special value: sinθ<-->θ<--> a&b parallel/ perpendicular
    - modulus 模的计算你会嘛--> √(x^2+y^2+z^2)
  - Area: St=1/2|a||b|sinθ= 1/2|a×b|--> beautiful as no need for angle
  - distributive and anticommutative
    - a×b= -b×a commute positions & 'minus sign'
  - component form
- 求夹角或相交线或brabra...: 由于我们只学了 parallel, perpendicular 这两种关系, 所以找关系然后找 fixed position 和 direction *
  - Shortest Distance formula: perpendicular relationship
    - a point and a plane: |PM|--> r(PM)= a + λ(n) and calculate its magnitude
    - a line and a plane
  - reflection of a point in a plane ??
  - intersection of many direction vectors in (2-D) planes--> 'multivariable functions' ??
  - line relationship
    - skewed or intercepted
      - perpendicular
      - intercepted point
        Gauss-jordan ellimination or the row reduction
    - parallel or coincident
  - plane relationship
    - intercepted
      - line of interception
        not unique but infinity answers, so one variable in the point coordination need to be defined by yourself
        Gauss-jorgan ellimination or using tool: Matrices--> The row reduction
    - parallel or coincident
      - from cartesian form Ax+By+Cz=D, if two have same A:B:C ratio, they are parallel. if they also have same D, they are coincident.
  - angle(θ): θ<=>cosθ =ab/(|a||b|) so we have + or - value to see weather it is acute or obtuse
    - between two planes: θ = θ(n)
    - between a line and a plane (line)
    - between normals and...
- Equation of a plan
  - parametric/ vector equation of a plane
    - r= a+λb+μc. λ,μ: scalar parameters, a: fixed position vector, b&c are noncollinear(nonparallel) direction vectors
  - cartesian equation of a plane
    - ax+by+cz=D is the cartesian equation of a plane perpendicular to n=( ), in aother form by identity(λ= ...= ...= ...) we can see the point in the plane & 'jumped ratio'
  - scalar product equation of a plane
    - (r-r1)·n=0--> rn=r1n--> rn=D (losing information)
- Equation of a line
  - X scalar equation
  - vector product equation
    - (r-a)×b=0
  - parametric/ vector equation
    - r=a+λb, λ: scalar parameter, a: fixed position vector, b: direction vector (parallel to the line)
  - cartesian equation