# Vector and Arrow in Space

This tutorial provides examples to explain vectors and arrows in space (vector visualization). It includes 2 and 3-dimension vectors as well as vector addition and subtraction.

## Example 1: 2 dimension vectors

$$\vec{V_1} =(3, 2 ) = \left[\begin{array}{ccc} 3\\ 2 \end{array} \right]$$

Note that, $$(3, 2 )$$ and $$\left[\begin{array} {ccc} 3\\ 2 \end{array} \right]$$ are equivalent. Further, $$\vec{V_1}$$ can be visualized as follows (i.e., plotted on the $$\mathbb{R}^2$$ ).

## Example 2: 3 dimension vectors

Of course, vectors can have more than 2 dimensions, such as 3 dimensions, or even more.

$$\vec{V_1} =(3, 2, 4 ) = \left[\begin{array}{ccc} 3\\ 2\\ 4 \end{array} \right]$$

The following is an example of vector addition and we use parallelogram law here.

$$\vec{Y}+\vec{X} = \left[\begin{array}{ccc} 1 \\ 3 \end{array} \right] + \left[\begin{array}{ccc} 3 \\ 2 \end{array} \right] =\left[\begin{array}{ccc} 4 \\ 5 \end{array} \right]$$

We can also use the triangle law for the addition of vectors. This method is also called the head-to-tail method. That is, two vectors can be added together by placing them together in such a way that the first vector’s head joins the tail of the second vector.

## Example 4: Vector subtraction

The vector subtraction of two vectors a and b is represented by a – b. For this, we can use the triangle law.

$$\vec{Y}-\vec{X} = \left[\begin{array}{ccc} 1 \\ 3 \end{array} \right] – \left[\begin{array}{ccc} 3 \\ 2 \end{array} \right]$$

It can be also written as adding the negative of vector b to the vector a. i.e., a – b = a + (-b). With that, we can use the parallelogram law.

$$\vec{Y}-\vec{X} =\vec{Y}+\vec{-X}= \left[\begin{array}{ccc} 1 \\ 3 \end{array} \right] + \left[\begin{array}{ccc} -3 \\ -2 \end{array} \right] =\left[\begin{array}{ccc} -2 \\ 1 \end{array} \right]$$