![]() It just depends on the order that you labeled the points, sometimes you're going to get a negative value when you subtract. If you look at the graph, you can see that the length of the vertical leg is a positive 6. When you subtract the y-coordinates you get -6. This will help you visualize how far apart they really are. Keep in mind that if the coordinates have different signs, it means they're on the opposite sides of the axis. Make sure to remember that subtracting is the same thing as adding the opposite. A curious and (hopefully) easy-to-understand example is the taxicab distance. It may seem an abstract useless thing, but it's tremendously useful: Einstein's relativity and other important sector of physics are built on this. More technical details can be found in Wikipedia's article about enter link description here. ) and if the Pythagorean Theorem (or some variation of it) holds. which lines can be called "straight", which are the triangles. So given - for example - a plane, one has first to define how the distance is computed and from this it will automatically follow the geometry of the plane (i.e. Nowadays, the definition of distance between two points deeply influences the geometry of the space you work in, and the euclidean distance (the one that corresponds to Euclidean Geometry) is just one of the (infinitely-)many possible choices. The contemporary way of thinking of distances is significantly different. So we can remove the dependency from #H#: Similarly, since #C# and #H# lay on the same vertical line, then #x_C=x_H#. Now, since #A# and #H# lay on the same horizontal line, then #y_A=y_H#. We got rid of the absolute values because squaring a number #z# or #-z# gives the same result. Using the Pythagorean Theorem, we can then compute the length hypotenuse #AC# i.e. So to compute the distance between #A# and #C# we can compute the distance between #A# and #H# (they lie on an horizontal line), and the distance between #C# and #H# (they lie on a vertical line). If we call #H# their intersection point, #ACH# is a right triangle and #AC# is the hypotenuse But we can trace an horizontal straight line through #A# and a vertical straight line through #C#. They belong to a straight line that it's neither horizontal nor vertical. In fact, let's think about computing the distance between the points #A# and #C# previously defined. This is quite easy, because the two points lie on the horizontal line #y=3#, so the distance between them can be interpreted as the length of the line segment #AB#, which is #5# and can be computed by subtracting the #x#-coordinates of the two points and considering the absolute value of the result (to avoid negative distances). a plane with a coordinate system such that).įirst of all, let's compute the distance between #A=(2,3)# and #B=(7,3)#. It's used to compute the distance between two points in an orthogonal coordinate system (i.e. The distance formula makes sense in a coordinate context. In other words, they are the same thing in two seemingly different contexts. In short, the distance formula is a formalization of the Pythagorean Theorem using #x# and #y# coordinates. (depending on if #x_1 > x_2# or #x_1 < x_2#, and similarly for #y#.) Or, we could put it another way through substitutions based on the distance definitions above. What do you see in these formulas? Have you ever tried drawing a triangle on a Cartesian coordinate system? If so, you should see that these are two formulas relating the diagonal distance on a right triangle that is composed of two component distances #x# and #y#. The greater the #x# contribution, the flatter the slope. The greater the #y# contribution, the steeper the slope. Any diagonal line segment has an #x# component and a #y# component, due to the fact that a slope is #Deltay"/"Deltax#.The distance from one point to another is the definition of a line segment.There is the relationship where #sqrt((x-c)^2) = color(green)(|x-c|) = x-c " AND " -x+c#.The definition of a distance from #x# to #pmc# is #color(green)(|x-c|)#.We commonly write the Pythagorean Theorem as:Ĭonsider the following major points (in Euclidean geometry on a Cartesian coordinate axis): The distance formula is commonly seen as: If we consider what the distance formula really tells you, we can see the similarities. ![]()
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