MA_12.1 12.2 12.3

12.1 Three Dimensional Coordinate
- Coordinate System
- Equation of a Sphere(球面方程)
12.2 Vectors
- Vectors
- Combining Vectors
- Components
- Vectors Addition and Scalar Multiplication
- Properties of Vectors
- Vectors
12.3 The Dot Product(Scalar/Inner Product)
- Physical background
- Properties of the dot product
- Dot Product

12.1 Three Dimensional Coordinate

Coordinate System

z=f(x,y),(x,y)∈Dz = f(x,y) , (x,y) \in Dz=f(x,y),(x,y)∈D

这表示一个二元函数，z由两个自变量x，y唯一确定，设在xoy平面上有一个区域D，则D为此二元函数的定义域，那么z=f(x,y)就确定了一个在oxyz空间直角坐标系内的一个曲面(平面算一种特殊的曲面)，定义域上任意一定均对应曲面上的一点，此点到xoy平面的距离即为z的绝对值。

The origin(原点)：A fixed point O
Coordinate axes(坐标轴): Three directed lines through O that are perpendicular(垂直的) to each other.

Right-hand rule(右手定则): The direction of the z axis is determined by the right-hand rule.

Coordinate plane(坐标平面)：The xy-plane is the plane that contains x- and y-axis,etc.
Octants(卦限): Eight octants.（三个相互垂直的平面划分空间的八个部分中的任意一个）

Coordinates(坐标): P(a, b, c)
Projections(投影) of P(a, b, c):

Projections of P onto xy-plane: Drop a perpendicular from P to the xy-plane, we get Q(a, b, 0) （投影到的平面缺哪一个坐标轴，投影点坐标哪一轴坐标为0） ,etc.

Equation of a Sphere(球面方程)

Distance Formula in Three dimensions:

The distance∣P1P2∣|P_1P_2|∣P1P2∣between the points P1(x1,y1,z1)P_1(x_1, y_1,z_1)P1(x1,y1,z1) and P2(x2,y2,z2)P_2(x_2, y_2,z_2)P2(x2,y2,z2) is

∣P1P2∣=(x1−x2)2+(y1−y2)2+(z1−z2)2|P_1P_2| = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2} ∣P1P2∣=(x1−x2)2+(y1−y2)2+(z1−z2)2

Equation of a Sphere:

An equation of a sphere with center C(h,k,l)C(h, k, l)C(h,k,l) and radius rrr is

(x−h)2+(y−k)2+(z−l)2=r2(x − h)^2 + (y − k)^2 + (z − l)^2 = r^2(x−h)2+(y−k)2+(z−l)2=r2

12.2 Vectors

Vectors

Vector:The term vector is used to indicate a quality that has both magnitude and direction.Displacement vector(位移): It has initial point A (the tail), the terminal point B (the tip), denote it by v⃗\vec{v}v = AB⃗\vec{AB}AB
Zero vector: 0⃗\vec{0}0 has length 0, no specific direction
Equivalent or equal: u⃗=v⃗\vec{u} = \vec{v}u=v means they have the same length and the same direction.

Combining Vectors

Definition of vector addition:If u⃗\vec{u}u and v⃗\vec{v}v are vectors positioned so the initial point of v⃗\vec{v}v is at the terminal point of u⃗\vec{u}u, then the sum u⃗\vec{u}u + v⃗\vec{v}v is the vector from the initial point of u⃗\vec{u}u to the terminal point of v⃗\vec{v}v.
Definition of scalar Multiplication（数乘）:If c is a scalar and v⃗\vec{v}v is a vector, then the scalar multiple cv⃗c\vec{v}cv is the vector which has the length ∣c∣⋅∣v⃗∣|c| · |\vec{v}|∣c∣⋅∣v∣ and has the same direction as v⃗\vec{v}v if c is positive and the opposite to u⃗\vec{u}u if c is negative.
Definition of difference:

u⃗−v⃗=u⃗+(−1)v⃗\vec{u} − \vec{v} = \vec{u} + (−1)\vec{v}u−v=u+(−1)v

Components

Treat vectors algebraically.

Representations of vector:

a⃗=(a1,a2),a⃗=(a1,a2,a3)\vec{a} = (a_1, a_2) , \vec{a} = (a_1, a_2, a_3)a=(a1,a2),a=(a1,a2,a3)
Position vector of the point P(a1,a2,a3)P(a_1, a_2, a_3)P(a1,a2,a3):

a⃗=OP⃗=(a1,a2,a3)\vec{a} = \vec{OP} = (a_1, a_2, a_3)a=OP=(a1,a2,a3)
The length of a⃗=(a1,a2,a3)\vec{a} = (a_1, a_2, a_3)a=(a1,a2,a3) is

∣a⃗∣=∣∣a⃗∣∣=a12+a22+a32|\vec{a}|=||\vec{a}||=\sqrt{a_1^2+a_2^2+a_3^2} ∣a∣=∣∣a∣∣=a12+a22+a32

Vectors Addition and Scalar Multiplication

If a⃗=(a1,a2)\vec{a} = (a_1 , a_2)a=(a1,a2) , b⃗=(b1,b2)\vec{b} = (b_1 , b_2)b=(b1,b2) and c is a number, then

a⃗+b⃗=(a1+b1,a2+b2)\vec{a} + \vec{b} = (a_1 + b_1 , a_2 + b_2)a+b=(a1+b1,a2+b2);
ca⃗=(ca1,ca2)c\vec{a} = (ca_1 , ca_2)ca=(ca1,ca2).

Properties of Vectors

If a⃗\vec{a}a , b⃗\vec{b}b and c⃗\vec{c}c are vectors in VnV_nVn , and c and d are scalars, then

a⃗+b⃗=b⃗+a⃗;a⃗+(b⃗+c⃗)=(a⃗+b⃗)+c⃗\vec{a} +\vec{b} = \vec{b} + \vec{a} ; \vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c}a+b=b+a;a+(b+c)=(a+b)+c (交换律、结合律)

a⃗+0⃗=a⃗;a⃗+(−a⃗)=0⃗\vec{a}+\vec{0}=\vec{a}; \vec{a}+(-\vec{a})=\vec{0}a+0=a;a+(−a)=0 !!!

c(a⃗+b⃗)=ca⃗+c⃗b⃗;(c+d)a⃗=ca⃗+da⃗c(\vec{a} +\vec{b}) = c\vec{a} + c⃗\vec{b} ;(c + d)\vec{a} = c\vec{a} + d\vec{a}c(a+b)=ca+c⃗b;(c+d)a=ca+da (分配律)

(cd)a⃗=c(da⃗),1a⃗=a⃗(cd)\vec{a} = c(d\vec{a}), 1\vec{a} = \vec{a}(cd)a=c(da),1a=a

Vectors

Let i⃗,j⃗\vec{i},\vec{j}i,j and k⃗\vec{k}k be the standard basis vectors(标准正交基) with the directions of positive x−x-x−, y−y-y−, and z−z-z−axes, respectively

i⃗=(1,0,0),j⃗=(0,1,0),k⃗=(0,0,1)\vec{i} = (1, 0, 0), \vec{j} = (0, 1, 0),\vec{k} = (0, 0, 1)i=(1,0,0),j=(0,1,0),k=(0,0,1)

Any vector a⃗=(a1,a2,a3)\vec{a} = (a_1, a_2, a_3)a=(a1,a2,a3) can be expressed in terms of i⃗,j⃗\vec{i},\vec{j}i,j and k⃗\vec{k}k. (重要！！！)

a⃗=a1i⃗+a2j⃗+a3k⃗.\vec{a} = a_1\vec{i} + a_2\vec{j} + a_3\vec{k}. a=a1i+a2j+a3k.

The unit vector（单位向量） that has the same direction as a⃗\vec{a}a with a⃗≠0⃗\vec{a} \ne \vec{0}a=0

u⃗=a⃗∣a⃗∣\vec{u}=\frac{\vec{a}}{|\vec{a}|}u=∣a∣a

12.3 The Dot Product(Scalar/Inner Product)

Physical background

Properties of the dot product

If a⃗,b⃗\vec{a},\vec{b}a,b and c⃗\vec{c}c are vectors in VnV_nVn , and c is a scalar, then

a⃗⋅a⃗=∣a⃗∣2;a⃗⋅b⃗=b⃗⋅a⃗\vec{a}\cdot \vec{a}=|\vec{a}|^2;\vec{a}\cdot \vec{b}=\vec{b}\cdot \vec{a}a⋅a=∣a∣2;a⋅b=b⋅a
a⃗⋅(b⃗+c⃗)=a⃗⋅b⃗+a⃗⋅c⃗;(ca⃗)⋅b⃗=c(a⃗⋅b⃗)=a⃗⋅(cb⃗)\vec{a}\cdot (\vec{b}+\vec{c})=\vec{a} \cdot \vec{b}+\vec{a} \cdot \vec{c};(c\vec{a})\cdot \vec{b}=c(\vec{a}\cdot \vec{b})=\vec{a}\cdot (c\vec{b})a⋅(b+c)=a⋅b+a⋅c;(ca)⋅b=c(a⋅b)=a⋅(cb)
0⃗⋅a⃗=0\vec{0}\cdot\vec{a}=00⋅a=0 !!!
a⃗⊥b⃗⇔a⃗⋅b⃗=0\vec{a}\bot\vec{b} \Leftrightarrow \vec{a} \cdot \vec{b}=0a⊥b⇔a⋅b=0

Dot Product

If θ is the angle between the vector a⃗\vec{a}a and b⃗\vec{b}b, then the dot product a⃗⋅b⃗\vec{a} \cdot \vec{b}a⋅b is a scalar defined by

a⃗⋅b⃗=∣a⃗∣∣b⃗∣cosθ,θ∈[0,π].\vec{a} · \vec{b} = |\vec{a}||\vec{b}| cos θ, θ ∈ [0, π].a⋅b=∣a∣∣b∣cosθ,θ∈[0,π].
If a⃗=(a1,a2)\vec{a} = (a_1 , a_2)a=(a1,a2) and b⃗=(b1,b2)\vec{b} = (b_1, b_2)b=(b1,b2), then

a⃗⋅b⃗=a1b1+a2b2.\vec{a} · \vec{b} = a_1b_1+a_2b_2.a⋅b=a1b1+a2b2.

Proof:
Generalization: Let a⃗=(a1,a2,⋅⋅⋅,an)\vec{a} = (a_1, a_2, · · · , a_n)a=(a1,a2,⋅⋅⋅,an) and b⃗=(b1,b2,⋅⋅⋅,bn)\vec{b} = (b_1, b_2, · · · , b_n)b=(b1,b2,⋅⋅⋅,bn) be two n-dimensional vectors, then the dot product between these two vectors can be defined as

a⃗⋅b⃗=a1b1+a2b2+⋅⋅⋅+anbn.\vec{a} \cdot \vec{b} = a_1b_1+ a_2b_2+ · · · + a_nb_n.a⋅b=a1b1+a2b2+⋅⋅⋅+anbn.
The direction angles(方向角) of a nonzero vector a⃗\vec{a}a are angles α, β and γ that a⃗\vec{a}a makes with positive x-, y-, and z-axes.
The direction cosines of vector a⃗\vec{a}a is

cosα=a⃗⋅i⃗∣a⃗∣=a1∣a⃗∣cos\alpha = \frac{\vec{a}\cdot\vec{i}}{|\vec{a}|}=\frac{a_1}{|\vec{a}|}cosα=∣a∣a⋅i=∣a∣a1

cosβ=a⃗⋅j⃗∣a⃗∣=a2∣a⃗∣cos\beta = \frac{\vec{a}\cdot\vec{j}}{|\vec{a}|}=\frac{a_2}{|\vec{a}|}cosβ=∣a∣a⋅j=∣a∣a2

cosγ=a⃗⋅k⃗∣a⃗∣=a3∣a⃗∣cos\gamma = \frac{\vec{a}\cdot\vec{k}}{|\vec{a}|}=\frac{a_3}{|\vec{a}|}cosγ=∣a∣a⋅k=∣a∣a3
(cos α, cos β, cos γ) is the unit vector(单位向量) in the direction of a⃗\vec{a}a. !!!
Two vectors a⃗\vec{a}a and b⃗\vec{b}b are perpendicular(垂直的) if and only if

a⃗⋅b⃗=0\vec{a} \cdot \vec{b}=0a⋅b=0
The scalar projection(标量投影) of b⃗\vec{b}b on a⃗\vec{a}a is

compa⃗b⃗=a⃗⋅b⃗∣a⃗∣comp_{\vec{a}} \vec{b} = \frac{\vec{a}\cdot\vec{b}}{|\vec{a}|}compab=∣a∣a⋅b
The vector projection(矢量投影) of b⃗\vec{b}b on a⃗\vec{a}a (a shadow of b⃗\vec{b}b ) is

proja⃗b⃗=(a⃗⋅b⃗∣a⃗∣)a⃗∣a⃗∣=a⃗⋅b⃗∣a⃗∣2a⃗proj_{\vec{a}} \vec{b} = ( \frac{\vec{a}\cdot\vec{b}}{|\vec{a}|})\frac{\vec{a}}{|\vec{a}|}= \frac{\vec{a}\cdot\vec{b}}{|\vec{a}|^2}\vec{a}projab=(∣a∣a⋅b)∣a∣a=∣a∣2a⋅ba

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工科数学分析 MA_12 Vectors and the Geometry of Space （上篇）

MA_12.1 12.2 12.3

12.1 Three Dimensional Coordinate

Coordinate System

Equation of a Sphere(球面方程)

12.2 Vectors

Vectors

Combining Vectors

Components

Vectors Addition and Scalar Multiplication

Properties of Vectors

Vectors

12.3 The Dot Product(Scalar/Inner Product)

Physical background

Properties of the dot product

Dot Product

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