Geometry

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1.Explain why we use the formula AB = |a - b| to find the distance of a segment. Include why you need to subtract and why you need to take the absolute value.



The formula for finding vector AB is AB = b - a where a and b are the position vectors OA and OB respectively

then the magnitude (length) of AB = |AB| = |b - a|

Whenever the absolute symbols | | are used it just means the magnitude ... so in the case of AB, |AB| means the magnitude (length) of AB and similarly for |b - a|

so like if it's A(3, 9) and B(-2, -6) then:

a = <3, 9> and b = <-2, -6>

then vector AB = b - a = <(-2 - 3), (-6 - 9)> = <-5, -15>

and the magnitude (length) of AB = |b - a| = √[(-5)² + (-15)²] = √[25 + 225] = √250 = 5√10


BUT notice that it is actually just an application of the usual distance formula b/c using the distance formula you get:

|AB| = √[(-2 - 3)² + (-6 - 9)²] = √[(-5)² + (-15)²] = √[25 + 225] = √250 = 5√10



BUT you could also use |AB| = |BA| = |a - b| b/c even tho AB and BA are in opposite directions the MAGNITUDE of AB and BA is the same ... BUT I wouldn't do it b/c AB ≠ a - b and you could mess the vectors up ... and so reversing the order ONLY works for the MAGNITUDE NOT the actual vector ...

anyway using the given formula |AB| = |a - b| to find the MAGNITUDE of AB you get:

|AB| = |<3, 9> - <-2, -6>| = |<(3 + 2), (9 + 6)>| = |<5, 15>| = √[25 + 225] = √250 = 5√10


so you get the same answer for the length of AB whichever method you use



btw ... the reason for subtraction is b/c of the vector triangle ... so to show why you subtract draw a vector triangle:

vector OA goes from the origin to point A ... mark it a

vector OB goes from the origin to point B ... mark it b

vector AB runs FROM A to B


OA + AB = OB

so a + AB = b

AB = b - a

and |AB| = |b - a|

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