Mathematics seems to throw letters our way on occasion, despite consisting primarily of numbers. We can forgive this in algebra, but when those letters seem to twist into strange symbols like the upside-down T, many of us feel lost all over again.
The upside-down T (or ⊥) refers to perpendicular lines, which are two lines that intersect at a 90-degree angle. They must be straight lines, and the point where they meet must also be at a right angle. If the lines are wavy or at any other angle, they are not perpendicular. In a geometric equation, you would represent perpendicular lines with ⊥.
This upside-down T symbol is relatively easy to understand and apply once you know what it represents and how to use it. So, to become a geometry wiz, read on.
What Are Perpendicular Lines?
Perpendicular lines cross or intersect and join at right angles (90-degrees). Therefore, you can use perpendicular lines for many mathematical equations and patterns, serving an essential purpose.
We see perpendicular lines all around us. Any square or rectangle has perpendicular corners. You can measure the corner of any door, and you should see perpendicular lines there, too.
The origin of the word also tells us its meaning. It comes from Old French (perpendiculer), which means forming from a right angle, which they derived from Latin (perpendiculum), meaning “plumb line” (source).
“Plumb line” refers to a straight or vertical line. Perpendicular lines do require straight lines, but one does not have to be vertical, so there has been some change in meaning.
Now, we can examine how you can use perpendicular lines in math.
What Does Perpendicular Mean in Math?
As we’ve stated above, the word “perpendicular” means the same thing in math.
Perpendicular (or orthogonal) lines cross over each other at a right angle. Thus, these lines must be straight and meet at a right (90-degree) angle. There are several ways that you can represent perpendicular lines (source). The smaller box represents the right angle in the diagram.
Lines R and S cross through each other at a right angle composed of perpendicular lines. In geometry, you would write it as R⊥S. In layman’s terms, this means that where R and S intersect, there is a right angle.
Perpendicular lines can meet, coming from any direction.
Even though the lines are meeting diagonally, their angle is still a right angle, and it would still be R⊥S. In these scenarios, perpendicular lines create a cross. Still, the edges of a square or rectangle would also be perpendicular lines.
In this scenario, all the corners are perpendicular lines as they meet at a 90-degree angle, and they would be A⊥B, B⊥C, C⊥D, and D⊥A.
Other shapes can also have perpendicular lines but not at all corners. An example of this would be a triangle, but it would have to look like a rectangle or square cut in half diagonally.
You would represent this as AB⊥BC. There are many other examples in shapes, but the main point is to have a right angle. The angles at A and C are not perpendicular as they are not 90-degree angles. We will discuss this further when examining Pythagoras Theorems.
One crucial point is that all perpendicular lines intersect (meet or cross over), but not all intersecting lines are perpendicular.
What Does ⊥ Mean in Linear Algebra?
Linear algebra ⊥ has a similar meaning, but you would apply it differently. When using it in linear algebra, mathematicians refer to it as orthogonality. The word comes from the Latin words “ortho,” meaning upright, and “gonia,” which means angle. Hence, its meaning is quite similar to perpendicular, but the focus is different.
Linear Algebra
Before understanding the role of ⊥, we need to understand what linear algebra is. Despite the name, it is not the same as the algebra you learned in school.
Linear algebra is the study of vectors and matrices, and many know it as the study of data. Linear combinations and Cartesian planes create information.
Linear Algebra uses numbers in columns called vectors — these include both magnitude and direction (source). Matrices use number arrays, consisting of a set of numbers in square brackets. We see ⊥ in Cartesian planes.
Cartesian Planes
When you look at the types of perpendicular lines in geometry, you will have seen that the lines crossing over one another looked like a Cartesian plane. Its two-dimensional form represents two lines that cross over one another.
A Cartesian plane is when two number lines intersect perpendicularly and form four quadrants. The horizontal line is the x-axis, and the vertical line is the y-axis (source).
Coordinates indicate the specific location of a point on a number line or plane. The middle point of the plane is 0, and the x-axis going towards the right shows positive numbers. When the x-axis moves towards the left, it indicates negative numbers.
Similarly, the y-axis moving up from zero shows positive numbers, and the y-axis line moving down from zero indicates negative numbers. When associating values to the Cartesian plane, the first number indicates the x-value, and the second number indicates the y-value.
Therefore, [-3, -3] would indicate that the point is in the lower-left quadrant while [-3, 1] would be in the upper-left quadrant.
All points on the Cartesian plane are orthogonal or perpendicular due to the nature of Cartesian planes because everything begins from point zero, which is the base of the right angle. So, you could express a point as 1, 1 ⊥ 0.
Pythagorean Theorem
Any self-respecting math student has likely encountered the theory of Pythagoras, but what does it have to do with an upside-down T?
When we examined perpendicular lines earlier, one of the shapes that came out was the triangle, where only one corner was perpendicular. The Pythagoras Theorem makes a relationship between the lengths of the three sides.
The theory of Pythagoras shows that in a triangle with a perpendicular corner, the length of the longest side is equal to the square root of the sum of the two other sides squared. Mathematically, you would state this as a2 + b2 = c2.
Because of the perpendicular side, it becomes easier to calculate how long each side of the triangle is. For example, imagine that you know the shorter sides of the triangle are three and four inches long, respectively.
Using the Pythagoras theorem, you would state: 32 + 42 = 25, making the longest side = 5. To break this down further, the square root of a number means to multiply by itself. So 3 x 3 = 9 and 4 x 4 = 16. 9 + 16 equals 25, which is the square root of 5.
What Does ⊥ Mean in Logic?
Even though ⊥ and 丄 look like lowercase and uppercase versions of the same symbols, they have significantly different meanings and usage.
The smaller ⊥ that we have been using means a perpendicular line. The bigger 丄, which logicians also call the up tack symbol, is only in logic and indicates that something is always false (source).
Examples
While truth values in mathematics and logic can be quite complicated, let’s look at a more straightforward example. Imagine that you love chocolate in all its forms. So the sentence “I love chocolate” would be an example of a truth value.
In contrast, if you said “I hate chocolate,” that would be an example of a false value, expressed with the up tack symbol: 丄.
Another example closer to mathematics would be the very simple equation: 1 + 1 = 2. Since this is true and there is no other correct answer, this is a truth value. In contrast, 1 + 1 = 3 is false, and no discussion would change that, making it 丄.
There are ways to distinguish between true and false statements. For example, the sentence “all married people are married” is entirely accurate due to the nature of its phrasing. If someone is not married, they cannot call themselves a married person.
When something is true in logic, you will write it with a capital T. You would express this thought as “All married people are married = T.”
However, the opinion “all married people are happy” cannot be valid. All married people cannot be happy, even if you are happy within your own marriage. You would represent this as “all married people are happy” = 丄.
Sometimes, logicians state that the 丄 symbol represents “falsum” and “absurdum.” Both are Latin terms that mean that something is entirely false and untrue.
Different Types of Lines
We have already broken down perpendicular lines, but other lines appear commonly in geometry.
There is a joke about lines in mathematics representing different kinds of love stories. Parallel lines are never meant to meet, tangent lines are lines that were once together but will never meet again, and asymptotes draw ever closer but will never meet.
Parallel lines are two lines that run adjacent to each other but do not touch or meet, no matter how long they get.
Next, we have tangent lines. Tangent lines appear in graphs with a curve. They show when a specific point touches the curve, but the arc bends away from the line immediately after that, and they do not meet again.
In the above diagram, the purple line is the tangent line, and it touches the data curve for only a short period. The red dot indicates this.
Finally, we have asymptote lines. In geometry, they appear on a Cartesian plane and work with the curve of a data line. The line continuously gets closer to the line, but it cannot ever touch the line.
In the figure above, the asymptote line (in red) closely follows the data curve but does not make contact with it.
These are not the only lines in geometry, of course, but just a few basic ones. In that sense, perpendicular lines are a positive love story. Two lines, coming from different directions, destined by fate to always meet at the “right” time.
Other T Symbols
We already know about the smaller ⊥ symbol indicating perpendicular lines and the larger 丄 up tack symbol. However, these are not the only symbols that use a T shape. Let us look at a few others.
In mathematics, a small t indicates the measurement of time. You can use it in a range of equations to show how many seconds, minutes, or hours have passed or need consideration.
For example, the formula for time is Time = Distance ÷ Speed. In shorthand, you would express it as t = d ÷ s. In a hypothetical situation, let us say we have to calculate how long it takes to walk 30 miles if you are walking at a speed of 5 miles an hour.
30 ÷ 5 = 6
T = 6 hours.
As previously indicated, a capital T tells the truth in a logical scenario compared to the larger 丄, which means false.
Finally, the ⫫ (double tack up) symbol is also in mathematics and logic. It represents an independent random variable. A random variable does not affect the calculation or experiment. Hence, you would not take it into account.
For example, if you were experimenting to find the best chocolate cake in the world, the number of eggs you used in each recipe would be an example of an independent random variable. While the eggs would affect the taste of the cake itself, it would not affect the taste of the other cakes.
Inverted Symbols in Math, Logic, and Geometry
In math, logic, and geometry, there are many symbols made up of letters, which are often upside down. Such symbols enable mathematicians and logicians to represent complex relationships in a simplified format.
To find out more about the upside-down A symbol, check out “What Does the upside-down A Mean in Math?” This article was written for strategiesforparents.com.
Letters are not the only things you can use for symbols — sometimes, you can also use numbers. For example, to find out more about the backward number 3, which we also call “epsilon,” read “What Does the Backwards 3 Symbol Mean?”
Final Thoughts
The ⊥ symbol is a vital part of geometry. When measuring shapes and calculating area, perpendicular lines tell us precisely what we are dealing with in a figure. While it is unlikely you will encounter the ⊥ symbol unless you are doing geometry, it is always good to know what you are looking for.