Geometry
posted by Unknown on .
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

Ask your teacher if you need help.