# 微积分作业代写calclulus代考|Some vector algebra essentials

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## 微积分作业代写calclulus代考|Unit vectors in 3-space

Let $a>0$ be a scalar, and let
\begin{aligned} \boldsymbol{v} &=(\alpha, \beta, \gamma) \ &=\alpha \mathbf{i}+\beta \mathbf{j}+\gamma \mathbf{k} \ &=\alpha \mathbf{e}{1}+\beta \mathbf{e}{2}+\gamma \mathbf{e}_{3} \end{aligned}
be a vector in $\mathbb{R}^{3}$ (see Section 1.B) with $x-y$, and $z$-components $\alpha, \beta$, and $\gamma$.

This vector has been written in the three most common forms appearing in current texts. The sets ${\mathbf{i}, \mathbf{j}, \mathbf{k}}$ and $\left{\mathbf{e}{1}, \mathbf{e}{2}, \mathbf{e}_{3}\right}$ represent the same set of unit vectors in mutually orthogonal directions in $\mathbb{R}^{3}$. The first form simply shows the components along the three orthogonal directions without reference to the unit vectors themselves, although the unit vectors and the coordinate system are implicit in this notation. The reader should be aware that we shall have occasion to refer to vectors using any of the three formats. The choice will depend on what is most convenient at that time without compromising understanding.

Multiplying a vector $v$ with a scalar will return a new vector with either the same direction if the scalar is positive or the opposite direction if the scalar is negative. In either case the resulting vector has different magnitude (Figure 1.1). This re-scaling will be a feature in Chapter 5 where we will need vectors of unit magnitude. For $a v$, with $a \in \mathbb{R}$, to be a unit vector we must have
$$|a \boldsymbol{v}|=|a||\boldsymbol{v}|=a \sqrt{\alpha^{2}+\beta^{2}+\gamma^{2}}=1, \text { i.e., } a=\frac{1}{\sqrt{\alpha^{2}+\beta^{2}+\gamma^{2}}} .$$
Therefore, to construct a unit vector in the direction of a specific vector $v$ we simply divide $\boldsymbol{v}$ by its length:
$$\boldsymbol{N}=\frac{\boldsymbol{v}}{|\boldsymbol{v}|}$$

## 微积分作业代写calclulus代考|The product of two vectors in 3-space

Let $\boldsymbol{u}$ and $\boldsymbol{v}$ be two non-parallel vectors in $\mathbb{R}^{3}$ :
$$\boldsymbol{u}=\left(a_{1}, a_{2}, a_{3}\right) \quad \boldsymbol{v}=\left(b_{1}, b_{2}, b_{3}\right) .$$
There are two particular product operations that we will utilize on many occasions. These are the vector and scalar products. From them very useful information can be extracted.
(a) A vector perpendicular to both $\boldsymbol{u}$ and $\boldsymbol{v}$ is
\begin{aligned} \boldsymbol{w}=\boldsymbol{u} \times \boldsymbol{v} &=\left|\begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_{1} & a_{2} & a_{3} \ b_{1} & b_{2} & b_{3} \end{array}\right| \ &=\left(a_{2} b_{3}-a_{3} b_{2}, a_{3} b_{1}-a_{1} b_{3}, a_{1} b_{2}-a_{2} b_{1}\right) \ &=-\boldsymbol{v} \times \boldsymbol{u} . \end{aligned}
This is called the “vector” or “cross” product. Note that $\boldsymbol{u} \times \boldsymbol{v}$ is antiparallel to $\boldsymbol{v} \times \boldsymbol{u}$. The relationship between the three vectors is shown in Figure $1.5$.
(b) The magnitude of the vector (cross) product of two vectors
$$|\boldsymbol{u} \times \boldsymbol{v}|=|\boldsymbol{w}|=|| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \ a_{1} & a_{2} & a_{3} \ b_{1} & b_{2} & b_{3} \end{array}||=\sqrt{\left(a_{2} b_{3}-a_{3} b_{2}\right)^{2}+\cdots}$$
gives the area of a plane parallelogram whose side lengths are $|\boldsymbol{u}|$ and $|\boldsymbol{v}|$

## 微积分作业代写calclulus代考|Unit vectors in 3-space

v=(一种,b,C) =一种一世+bj+C到 =一种和1+b和2+C和3

|一种v|=|一种||v|=一种一种2+b2+C2=1, IE， 一种=1一种2+b2+C2.

ñ=v|v|

## 微积分作业代写calclulus代考|The product of two vectors in 3-space

(a) 垂直于两者的向量你和v是

(b) 两个向量的向量（叉）积的大小
|你×v|=|在|=||一世j到 一种1一种2一种3 b1b2b3||=(一种2b3−一种3b2)2+⋯