### 物理代写|理论力学作业代写Theoretical Mechanics代考|Elementary Mathematical Operations

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## 物理代写|理论力学作业代写Theoretical Mechanics代考|Elementary Mathematical Operations

Two vectors $\mathbf{a}$ and $\mathbf{b}$ are added by a parallel translation of one of the vectors, say $\mathbf{b}$, such that the base point of $\mathbf{b}$ coincides with the arrowhead of the other vector a (Fig. 1.36). The sum vector $(\mathbf{a}+\mathbf{b})$ then starts at the base point of $\mathbf{a}$ and goes to the arrowhead of $\mathbf{b}$. One recognizes that $(\mathbf{a}+\mathbf{b})$ corresponds to the diagonal of the parallelogram spanned by $\mathbf{a}$ and $\mathbf{b}$ (parallelogram law). We list up some obvious rules for vector sums:
$(\alpha)$ Commutativity
$$\mathbf{a}+\mathbf{b}=\mathbf{b}+\mathbf{a}$$
This follows directly from the definition of the sum vector and becomes immediately clear with Fig. 1.37. Decisive for the commutativity is the free parallel mobility of the vectors in the plane.
( $\boldsymbol{\beta}$ ) Associativity
$$(\mathbf{a}+\mathbf{b})+\mathbf{c}=\mathbf{a}+(\mathbf{b}+\mathbf{c})$$

The validity of (1.130) can easily be read off from Fig. 1.38.
( $\gamma$ ) Vector Subtraction
$$\mathbf{a}-\mathbf{b}=\mathbf{a}+(-\mathbf{b})$$
Subtracting a from itself yields the so-called
$$\text { zero (null) vector: } \quad 0=a-a$$
the only vector which has no definite direction (Fig. 1.39). For all vectors holds:
$$\mathbf{a}+\mathbf{0}=\mathbf{a}$$
Because of (1.129), (1.130), (1.132) and (1.133) the set of all position vectors build a (commutative) group.
(b) Multiplication by a (Real) Number
Let $\alpha$ be a real number $(\alpha \in \mathbb{R})$ and $\mathbf{a}$ be an arbitrary vector.

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Scalar Product

As scalar (inner, dot) product of two vectors $\mathbf{a}$ and $\mathbf{b}$ is denoted by the following number (scalar):
$$(\mathbf{a}, \mathbf{b}) \equiv \mathbf{a} \cdot \mathbf{b}=a b \cos \vartheta, \quad \vartheta=\varangle(\mathbf{a}, \mathbf{b})$$
Illustratively, it is the product of the length of the second vector with the projection of the first vector on the direction of the second (see Fig. 1.41).
$$\begin{gathered} \mathbf{a} \cdot \mathbf{b}=0, \text { if 1) } a=0 \text { or/and } b=0 \ \text { or 2) } \vartheta=\pi / 2 . \end{gathered}$$

$\mathbf{a}$ and $\mathbf{b}$ are orthogonal $(\mathbf{a} \perp \mathbf{b})$ if
$$\mathbf{a} \cdot \mathbf{b}=0 \quad \text { with } a \neq 0 \text { and } b \neq 0 .$$
Properties
(a) Commutativity
$$\mathbf{a} \cdot \mathbf{b}=\mathbf{b} \cdot \mathbf{a}$$
This relation is directly perceptible from the definition of the scalar product.
(b) Distributivity
$$(\mathbf{a}+\mathbf{b}) \cdot \mathbf{c}=\mathbf{a} \cdot \mathbf{c}+\mathbf{b} \cdot \mathbf{c}$$
Figure $1.42$ gives immediately the proof, which again exploits the free relocatability of the vectors in the plane.
(c) Bilinearity (Homogeneity)
For each real number $\alpha$ holds:
$$(\alpha \mathbf{a}) \cdot \mathbf{b}=\mathbf{a} \cdot(\alpha \mathbf{b})=\alpha(\mathbf{a} \cdot \mathbf{b})$$
Proof (Fig. 1.43)
\begin{aligned} \alpha>0: \quad(\alpha \mathbf{a}) \cdot \mathbf{b} &=\alpha a b \cos \vartheta \ \mathbf{a} \cdot \mathbf{b} &=a b \cos \vartheta \end{aligned}

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Vector (Outer, Cross) Product

The product discussed in the last section assigns a number, i.e. a scalar, to the product of two vectors of a vector space. However, there exists a second type of product which addresses to two vectors a third vector from the same vector space. This is known as vector product, outer product, or cross product
$$\mathbf{c}=\mathbf{a} \times \mathbf{b}$$
This vector has the following properties:
$1 .$
$$c=a b \sin \theta ; \quad \vartheta=\varepsilon(\mathbf{a}, \mathbf{b})$$
The magnitude $c$ of the resulting vector corresponds to the area of the parallelogram spanned by the vectors $\mathbf{a}$ and $\mathbf{b}$ (Fig. 1.45).

1. $\mathbf{c}$ is oriented perpendicular to the area defined by $\mathbf{a}$ and $\mathbf{b}$ in such a way that $\mathbf{a}, \mathbf{b}$, $\mathbf{c}$ in this sequence build a right-handed coordinate system.

The second point indicates that the vector product does not simply characterize a direction but more a ‘direction of rotation, rotation sense’. Thus, in various respects the properties of a vector product are different from those of a ‘ordinary’ (polar) vector. $\mathbf{c}$ is a so-called axial vector (pseudovector). The strict distinction becomes clear with the term
Space Inversion
Reflection of all space points $\left(E_{3}\right)$ with respect to a fixed, given point, e.g. the origin of coordinates.

Polar vectors change their signs by inversion (see Fig. 1.46). On the other hand, since the rotation sense does not change after inversion, the axial vector will not change its sign (see Fig. 1.47).

We add a remark. It is clear that the scalar product of either only polar vectors or only axial vectors does not change its sign with inversion, being therefore a genuine scalar. The scalar product of a polar and an axial vector, however, changes into its negative and is for this reason called a pseudoscalar.

One has to bear in mind that the scalar product (Sect. 1.3.2) is defined between vectors of an arbitrary-dimensional vector space, while the vector product holds only for three-dimensional vectors.

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Elementary Mathematical Operations

(一种)交换性

(b) 关联性

(一种+b)+C=一种+(b+C)

(1.130) 的有效性可以很容易地从图 1.38 中读出。
(C) 向量减法

零（空）向量： 0=一种−一种

(b) 乘以一个（实数）数
Let一种是一个实数(一种∈R)和一种是任意向量。

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Scalar Product

(一种,b)≡一种⋅b=一种b因⁡ϑ,ϑ=\变量(一种,b)

(a) 交换性

(b) 分配性

(一种+b)⋅C=一种⋅C+b⋅C

(c) 双线性（同质性）

(一种一种)⋅b=一种⋅(一种b)=一种(一种⋅b)

## 物理代写|理论力学作业代写Theoretical Mechanics代考|Vector (Outer, Cross) Product

C=一种×b

1.

C=一种b罪⁡θ;ϑ=e(一种,b)

1. C垂直于由定义的区域定向一种和b以这样的方式一种,b, C在这个序列中建立一个右手坐标系。

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