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# Lesson 9.2: Vectors

## Symbols

$⇀\phantom{\rule{.3em}{0ex}}\text{single-barbed right arrow}$
⠳⠈⠗⠕

$\text{expression directly above}$
⠨⠔

$\text{bold symbol indicator}$
⠘⠆

$\text{bar under previous item}$
⠠⠱

$\text{bar over previous item}$

$\text{single right-pointing arrow over previous item}$
⠘⠱

## Review

The scope of a level change indicator is the next item. Braille grouping indicators must be used to enclose an expression that includes more than one item. An arrow is considered one item and does not require grouping symbols.

When complex algebraic equations contain indicators such as superscript, subscript, fractions, radicals, and/or letters standing alone, it is best to enclose them in grade 1 passage indicators to ensure the symbols are well defined without the need for grade 1 symbol indicators. When a complex expression is comprised of a single symbol sequence, a grade 1 word indicator will be enough to ensure that the various indicators are well defined without the need for grade 1 symbol indicators. The arrow indicator must be in grade 1 mode because it also has a grade 2 meaning.

In braille, each row of a matrix begins with the enlarged opening grouping symbol and ends with the enlarged closing grouping symbol. The enlarged grouping symbols must be aligned vertically, with the columns left adjusted.

## Explanation

A vector is an object having direction as well as magnitude, especially in determining the position of one point in space relative to another. Vectors are written in a number of ways. The most common way is to write the letters of the vectors head and tail with an arrow directly above the letters. The type of arrow used in this notation can be a single-barbed arrow with the barb to the right and on top of the shaft, or a standard-right pointing arrow with a full barb. The most common usage is the single barbed arrow. This requires the use of the expression directly above indicator followed by the symbols that form the arrow. A single-barbed arrow with the barb to the right and on top of the shaft uses four cells. The arrow indicator in cell one is dots one two five six; the regular barb, upper half, in line is dot four, dots one two three five; and the closing indicator to indicate the line of direction is dots one three five. The arrow indicator requires grade 1 mode.

When the vector is written with a standard right-pointing arrow above the letters, the simple right-pointing arrow directly above previous item (dots four five, dots one five six) can be used in braille.

Vectors are also frequently denoted by boldface type, with or without the arrow above. Boldface is only used in braille if it is the only method used to identify the vector.

Vectors can be written within a single row or single column matrix, called row vectors or column vectors. Follow formatting rules for matrices and use the appropriate enlarged grouping signs.

### Example 1

$\stackrel{⇀}{c}$
⠰⠰⠉⠨⠔⠳⠈⠗⠕

In Example 2, the letters appear in boldface type with the single barbed arrow above them. Boldface indicators are not used in braille because the arrows are also present.

### Example 2

$\stackrel{⇀}{\mathbf{o}}+\stackrel{⇀}{\mathbf{p}}$
⠰⠰⠕⠨⠔⠳⠈⠗⠕⠐⠖⠏⠨⠔⠳⠈⠗⠕

In Example 3, the letter representing the vector appears in boldface type. Since boldface is the only method used to identify the letter as a vector, the bold symbol indicator is used in braille.

### Example 3

$\mathbf{a}=\left(7,3\right)$
⠘⠆⠁⠀⠐⠶⠀⠐⠣⠼⠛⠂⠀⠼⠉⠐⠜

### Example 4

$\stackrel{⇀}{v}=\sqrt{{a}^{2}+{b}^{2}}$
⠰⠰⠰⠧⠨⠔⠳⠈⠗⠕⠀⠐⠶⠀⠩⠁⠔⠼⠃⠐⠖⠃⠔⠼⠃⠬⠰⠄

### Example 5

$\stackrel{⇀}{p}=\frac{2}{3}\left(\stackrel{⇀}{r}-\stackrel{⇀}{t}\right)$
⠰⠰⠰⠏⠨⠔⠳⠈⠗⠕⠀⠐⠶⠀⠼⠃⠌⠉⠐⠣⠗⠨⠔⠳⠈⠗⠕⠐⠤⠞⠨⠔⠳⠈⠗⠕⠐⠜⠰⠄

In Examples 6 through 8 the only symbols requiring grade 1 mode are on the left side of the equation. Grade 1 word indicators are used.

### Example 6

$\stackrel{⇀}{v}=\left(5,0\right)$
⠰⠰⠧⠨⠔⠳⠈⠗⠕⠀⠐⠶⠀⠐⠣⠼⠑⠂⠀⠼⠚⠐⠜

### Example 7

$\stackrel{⇀}{v}=\left[\begin{array}{c}5\\ 0\end{array}\right]$
⠰⠰⠧⠨⠔⠳⠈⠗⠕⠀⠐⠶⠀⠠⠨⠣⠼⠑⠠⠨⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⠣⠼⠚⠠⠨⠜

### Example 8

$\stackrel{⇀}{a}=\left[\begin{array}{c}6\\ -2\end{array}\right]$
⠰⠰⠁⠨⠔⠳⠈⠗⠕⠀⠐⠶⠀⠠⠨⠣⠀⠀⠼⠋⠠⠨⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠨⠣⠐⠤⠼⠃⠠⠨⠜