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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.

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