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.
We use the formula AB = |a - b| to find the distance of a segment because it is derived from the concept of vectors.
In this formula, a and b are the position vectors OA and OB, respectively. The position vector represents the direction and magnitude of a vector from the origin to a specific point.
To find the vector AB, we subtract vector a (OA) from vector b (OB). This subtraction gives us a new vector that represents the displacement from point A to point B.
Now, to find the magnitude (length) of AB, we calculate |AB|, which is the magnitude of the vector AB. Taking the absolute value ensures that the result is always positive, regardless of the direction of the vector.
In the formula |AB| = |a - b|, we're essentially finding the magnitude of the vector difference between a and b, which gives us the length of the segment AB.
By substituting the coordinates of the points A and B into the formula, we can find |AB| by performing the necessary calculations or using the distance formula. The distance formula is derived from the Pythagorean theorem and is commonly used to find the distance between two points in a coordinate system, such as in the example shared.
Therefore, by understanding the concept of vectors and how to calculate their magnitudes, we can use the formula AB = |a - b| to find the distance of a segment.