White H House of Math logo on multicolored square
Colorful House of Math logo
Log in

Finding the Distance Between Two Points Using Vectors

A vector from (8, 12) to (16, 18)

The distance between two points P = (x1,y1) and Q = (x2,y2) is the length of the vector PQ→. This formula finds the distance between the two points (the length of the vector) for you:

Formula

The Distance Between Two Points

|PQ→| = (x2 − x1 ) 2 + (y2 − y1 ) 2

Example 1

Find the distance between A = (8, 12) and B = (16, 18).

|AB→| = (16 − 8 ) 2 + (18 − 12 ) 2 = 82 + 62 = 64 + 36 = 100 = 10

Example 2

Decide s such that |AB→| = 4, when A = (s, 1) and B = (2, 5)

Here, you can insert the numbers right into the formula and solve for s:

|AB→| = (2 − s ) 2 + (5 − 1 ) 2 = 4 (2 − s ) 2 + (4 ) 2 = 4 (2 − s)2 + 16 = 16 4 − 4s + s2 = 0

Solving the equation s2 − 4s + 4 = 0:

s = 4 ±16 − 4 ⋅ 1 ⋅ 4 2 = 4 2 = 2.

Because this is a radical equation, you need to test your answer:

The left-hand side is

(2 − 2) 2 + (5 − 1) 2 = 16 = 4,

and the right-hand side is 4. As the left-hand and right-hand side are equal, s = 2 makes the length |AB→| equal to 4.

Want to know more?Sign UpIt's free!
White arrow pointing to the LeftPrevious entry
What Is the Length of a Vector?
Go back