Space vector

You may have already encountered a vector represented by a directed line segment in a plane or space. Here, the vector represented by the directed line segment in the three-dimensional space is called space vector．A vector in space represents a line segment with direction and magnitude.

From now on, the vector of space will always be represented by a directed line segment. It can be said that the vectors of space are equal as long as they have the same direction and magnitude, unlike the numbers you have dealt with so far. In other words, any vector in space can be replaced by a vector in space that is parallel to it and has the same magnitude, and from this, the vector in space can move freely without changing its direction and magnitude.．

First, we define two elementary operations, vector addition and scalar multiplication, for a vector in space. The operation of creating the sum of vectors is called addition, and multiplying the vector by a constant is called scalar multiplication.

When orthogonal coordinates are prepared for the space vector $\boldsymbol{x}$ and the origin and end points are translated so that the starting point is the origin. The vector connecting the origin and end coordinates $(x_{1},x_{2},x_{3})$ is $\boldsymbol{x} = \left [\begin{array}{c} x_{1} \\ x_{2} \\ x_{3} \end{array} \right]$ or expressed as ${}^t [x_{1}\ x_{2}\ x_{3}]$．Especially when dealing with $xyz$-space, a vector created by connecting points $(1,0,0),(0,1,0),(0,0,1)$ with the origin as the starting point. Using $\boldsymbol{i}, \boldsymbol{j}, \ boldsymbol{k}$, ${\ bf x} = x_{1} \boldsymbol{i} + x_{2} \boldsymbol{j} + x_{3}\ boldsymbol{k}$．Then the sum of two vectors $\boldsymbol{x} = x_{1}\:\boldsymbol{i} + x_{2}\:\boldsymbol{j} + x_{3}\:\boldsymbol{k}$， ${\mathbf y} = y_{1}\:\boldsymbol{i}+ y_{2}\:\boldsymbol{j} + y_{3}\:\boldsymbol{k}$ is defined by

$$x_{1}\:\boldsymbol{i} + x_{2}\:\boldsymbol{j} + x_{3}\:\boldsymbo... ...:\boldsymbol{i} + (x_{2}+y_{2})\:\boldsymbol{j} + (x_{3} + y_{3})\boldsymbol{k}$$

Also, the magnitude of the vector $\vert x_{1}\: \boldsymbol{i} + x_{2} \: \boldsymbol{j} + A_{3} \: \boldsymbol{k}\vert$ is the distance of the line segment connecting origin and point $(x_{1},x_{2},x_{3})$, which is represented by $\sqrt{x_{1}^{2} + x_{2}^{2} + x_{3}^{2}}$.

Example 1..1  

For $\boldsymbol{A} = -\boldsymbol{i} + 2\:\boldsymbol{j} + 3\:\boldsymbol{k}, \boldsymbol{B} = -3\:\boldsymbol{j} + 2\:\boldsymbol{k}$, find the followings:．

$$(a) \ 2\boldsymbol{A} \hskip 1cm (b) \ 3\boldsymbol{B} - 2\boldsymbol{A} \hskip 1cm (c)\ \vert\boldsymbol{A}\vert$$

Answer (a) $2\boldsymbol{A} = 2(-\boldsymbol{i} + 2\:\boldsymbol{j} + 3\:\boldsymbol{k}) = -2\:\boldsymbol{i} + 4\:\boldsymbol{j} + 6\:\boldsymbol{k}$.
(b) $3\boldsymbol{B} - 2\boldsymbol{A} = 3(-3\:\boldsymbol{j} + 2\:\boldsymbol{k}) -... ...+ 2\:\boldsymbol{j} + 3\:\boldsymbol{k} = 2\:\boldsymbol{i} -13\:\boldsymbol{j}$.
(c) $\vert\boldsymbol{A}\vert = \sqrt{(-1)^2 + 2^2 + 3^2} = \sqrt{14}.$

Question 1..1  

Find $\vert\boldsymbol{A} + \boldsymbol{B}\vert$

Basic vector

Vector $\: \boldsymbol{i},\: \boldsymbol{j},\: \boldsymbol{k}$ formed by connecting the origin and the point $(1,0,0),(0,1,0),(0,0,1)$ is called the basic vector in the Cartesian coordinate system $(O-xyz)$. Also, a vector with a magnitude of 1 such as $\boldsymbol{i}, \boldsymbol{j}, \ boldsymbol{k}$ is called a unit vector.