# Video: Using Vector Subtraction and Pythagorean Theorem to Find the Distance between Two Points in Cartesian Coordinates

Two points in the Cartesian plane have the coordinates (2.00 m, β4.00 m) and (β3.00 m, 3.00 m). Find the distance between them.

02:43

### Video Transcript

Two points in the Cartesian plane have the coordinates 2.00 meters, negative 4.00 meters and negative 3.00 meters, 3.00 meters. Find the distance between them.

Letβs start by drawing a sketch of these two points on the Cartesian plane. On these axes, each tick mark represents a distance of 1.00 meters. The first point mentioned has an π₯-value of 2.00 meters then a π¦-value of negative 4.00 meters; thatβs here on the plane.

The second point is at negative 3.00 meters plus 3.00 meters; thatβs here. Weβre asked to solve for the distance between these two points. That distance which weβll call π is the length of the straight line path from one point to the other.

If we draw out the change in π₯ position from one point to another, calling that π₯π₯, and the change in π¦ position between the two points, calling that π₯π¦, we can see that we formed a right triangle and π is the hypotenuse of that triangle.

Solving for distance in general between points on a Cartesian plane, that distance π is equal to the square root of π₯π₯ squared plus π₯π¦ squared. Looking at the two points that weβve been given, letβs call the first point π sub one and the second point π sub two.

If we apply the distance relationship to our two points π one and π two, then we can rewrite π₯π₯ and π₯π¦ as the difference between the π₯- and π¦-coordinates of π one and π two, respectively.

π₯π₯ is equal to 2.00 minus minus 3.00 meters or 5.00 meters. π₯π¦ equals negative 4.00 meters minus 3.00 meters or negative 7.00 meters. When we square π₯π₯ and π₯π¦, add them together, and take their square root, we find a total distance of 8.60 meters. Thatβs how far point one and point two are away from one another.