Geometric vector and Vector space

We have seen a vector in the coordinate plane represented by arrow head. In this section, to distinguish a general vector from a arrow headed vector, we call an arrow headed vector geometric vector. A geometric vector has a direction and a size. We have not defined a direction and a size. So, the definition of a geometric vector is not complete.

From now on, a geometric vector is represented by the arrow headed vector. A geometric vector is different from the number you used to. One thing is that if two geometric vectors have the same direction and the same magnitude, then these two geometric vectors are treated as the same vectors. Thus, any two geometric vectors are same if they have the same direction and magnitude. So, we can move a geometric vector freely without changing the direction and the magnitude.

Now we define two basic operation on the geometric vectors. First one is the vector addition and the second one is the scalar multiplication.

$\spadesuit$Addition of geometric vectors $\spadesuit$

To add two geometric vectors, we put the tail of one goemetric vector to the tip of another geomteric vector. Thus, the addition of two geometric vectors is defined as the geometric vector connecting the tail of one geometric vector and the rip of another geometric vector. In other words, it is a diagonal of th eparallelogram.

Figure 1.1: Addition of geometric vectors and scalar multiplication

The properties of the addition of geometric vectors are as follows:

1.

An addition of geometric vectors is a geometric vector. (closure)

2.

For any geometric vectors A and B, A+B = B+A . (commutative)

3.

For any geometric vectors A,B,C, (A+B)+C = A+(B+C). (associative)

4.

For any geometric vector A, there exists 0 such that A+0 = A (existence of zero element)

5.

For any geometric vector A, there exists B such that A+B = 0. (existence of inverse element)

$\spadesuit$A scalar multiple of geometric vectors $\spadesuit$

A scalar multiple of a geometric vector $\alpha$ A is defiend by the same direction of A and the magnitudeh is $\alpha$ times. If $\alpha$ is negative, then the direction of $\alpha$A is opposite to the original direction.

6.

A scalar multiple of a geometric vector is again a geometric vector.

7.

For any real numbers $\alpha$ and $\beta$, $\alpha$($\beta$A) = ( $\alpha\beta$)A. (associative law)

8.

For any real number $\alpha$ and $\beta$ , ( $\alpha + \beta$)A = $\alpha$A + $\beta$A, and for any geometric vectors A and B , $\alpha$(A+B) = $\alpha$A + $\alpha$B(distributive law)

9.

1A = A; 0A = 0; $\alpha$0 = 0

$\spadesuit$Geometric vector space $\spadesuit$

For any geometric vectors in coordinate plane or sapce, the properties 1 thru 9 holds. When the properties 1 thru 9 are satisfied, we call this set of geometric vectors geometric vector space. A geometric vector space is often called Euclid space and this is a typical example of abstract vector space. In general, a set under an addition and a scalar multiple satisfies the properties 1 thru 9, it is called a vector space. We often call an element in a vector space simply a vector. So, in this sense, you may see a lot of vectors not like geometric vectors.

$\spadesuit$Vectors in Space $\spadesuit$

For any geometric vector ${\mathbf x}$ on a coordinate plane, put the tail of a geomeric vector to the origin. Then the coordinate of the head of the vector is $(x_{1}, x_{2})$. Then ${\mathbf x} = (x_{1}, x_{2})$ represents the vector. Thus, the addition of two geometric vectors ${\mathbf x} = (x_{1}, y_{1}), {\mathbf y} = (y_{1},y_{2})$ can be defined by、

$$(x_{1},x_{2}) + (y_{1},y_{2}) = (x_{1}+y_{1}, x_{2}+y_{2})$$

Example 1..1  

For ${\mathbf X} = (-1,2), {\mathbf Y} = (0,-3)$, find ${\mathbf X} + {\mathbf Y}$.

Answer ${\mathbf X} + {\mathbf Y} = (-1,2) + (0,-3) = (-1,-1)$ $\blacksquare$

To generaloze this idea, we provide the geometric vectors in three dimension. The move the tail of the geometric vector to $(0,0,0)$. Then assume the coordinate of the head of the vector is $(a_{1},a_{2},a_{3})$.

Space vector is represented by $(a_{1},a_{2},a_{3})$. $a_{1}$ is called $x$-component, $a_{2}$ is $y$-component, $a_{3}$ is $z$-component. The set of these vectors are represented by { $(a_{1},a_{2},a_{3}) : a_{i}$ real} , An addtion and scalar multiplication are defined in the way so that this set can become a vector space. This vector space is called three dimensional vector space and denoted by ${ R}^3$ . In ${ R}^3$ , addition of these vectors are defined as follows:

$$(a_{1},a_{2},a_{3}) + (b_{1},b_{2},b_{3}) = (a_{1}+b_{1}, a_{2}+b_{2}, a_{3}+b_{3}).$$

An scalar multiplication of these vectors are defined as follows:

$$\alpha(a_{1},a_{2},a_{3}) = (\alpha a_{1}, \alpha a_{2}, \alpha a_{3}).$$

The zero element of this vector space is called zero vectorand denoted by $(0,0,0)$. 3-tuples $(a_{1},a_{2},a_{3})$ has no direction and magnitude. Once a 3-tuple $(a_{1}, a_{2}$, $a_{3})$ is considered as a point $(a_{1},a_{2},a_{3})$ on the coordinate space. Then we can think of this as a directed line starting from the origin. With this idea, we can think of a space vector as a geometric vector. Then the magnitude of a space vector $\Vert(a_{1},a_{2},a_{3})\Vert$ is the distance between $(a_{1},a_{2},a_{3})$ and the origin. Thus, we can write $(a_{1}^{2}+a_{2}^{2}+a_{3}^{2})^{1/2}$.

Example 1..2  

For ${\bf A} = (-1,2,3), {\bf B} = (0,-3,2)$, find the followings:
$(a) 2{\bf A} (b) 3{\bf B} - 2{\bf A} (c) \Vert{\bf A}\Vert (d) \Vert{\bf A} + {\bf B}\Vert$

Answer (a) $2{\bf A} = 2(-1,2,3) = (-2,4,6)$.
(b) $3{\bf B} - 2{\bf A} = 3(0,-3,2) -2(-1,2,3) = (2,-13,0)$.
(c) $\Vert{\bf A}\Vert = ((-1)^2 + 2^2 + 3^2)^{1/2} = \sqrt{14}.$
(d) $\Vert{\bf A} + {\bf B}\Vert = \Vert(-1,2,3)+(0,-3,2)\Vert = \Vert(-1,-1,5)\Vert = \sqrt{27} = 3\sqrt{3}.$ $\blacksquare$

(1,0,0),(0,1,0),(0,0,1) can be expressed by i,j,k. Using this symbol, we can write ${\bf A} = (a_{1},a_{2},a_{3})$ as $a_{1}{\bf i} + a_{2}{\bf j} + a_{3}{\bf k}$ . The vectors like i,j,k whose magnitude is 1 is called unit vector.

We can extend the idea about 3-tuple to n-tuples. But for the sake of the space of paper, 5-tuple is represented by $\left (\begin{array}{c} a_{1} \\ a_{2} \\ a_{3} \\ a_{4} \\ a_{5} \end{array} \right )$ insted of $(a_{1},a_{2},a_{3},a_{4},a_{5})$.

The addition and the scalar multiplication are defined as follows:

$$\left ( \begin{array}{c} a_{1} \\ a_{2} \\ a_{3} \\ a_{4} \\ ... ...2} \\ a_{3} + b_{3} \\ a_{4} + b_{4} \\ a_{5} + b_{5} \end{array} \right ),$$

$$\alpha \left (\begin{array}{c} a_{1} \\ a_{2} \\ a_{3} \\ a_{4... ... \\ \alpha a_{3} \\ \alpha a_{4} \\ \alpha a_{5} \end{array} \right ).$$

Similarly, we can defined an addtion and a scalar multiplication for $n$-tuples. Furthermore, these vectors satisfy the properties 1 thru 9.

Theorem 1..1  

$R^{n} = \{(a_{1},a_{2},\ldots,a_{n}) : a_{i} \in R\}$ is a vector space.

Exercise1−2

1. For vectors A = $(2,-1,3)$ and B = $(-1,1,4)$, find the followings:

(a) ${\bf A} + {\bf B}$ (b) $3{\bf B}$ (c) $\Vert{\bf A}\Vert$ (d) $3{\bf B} - 2{\bf A}$

2. For vectors A = $(-2,-1,3)$ and B = $(3,0,1)$, express the following vectors using ${\bf i},{\bf j},{\bf k}$ .

(a) ${\bf B} - {\bf A}$ (b) $3{\bf A} + {\bf B}$ (c) $3{\bf B} - 2{\bf A}$ (d) $${\frac{{\bf A} + {\bf B}}{2}}$$

3. Show that geometric vectors defined in 1.1 satifies the properties 1 thru 5.

4. Show that geometric vectors defined in 1.1 satifies the properties 6 thru 9.

5. Show that geometric vectors defined in 1.1 satifies the properties 1 thru 9. . 6. Two vectors ${\bf A}$ and ${\bf B}$ have the same initial point. Find the vector ${\bf C}$ which disect the angle between ${\bf A}$ and ${\bf B}$. 7. Solve the equation $x(1,1,1) + y(1,1,0) + z(1,0,0) = (2,-3,4)$. Find $x, y, z$.