Question

If A=(-2,-4) and B=(-8,4) what is the length of AB

Answer 1

The length of AB is
`10`

Explanation:

Given two points in the plane

`A=(a1,a2)` and
`B=(b1,b2)`

The distance between this two points is

`d(A,B)=\sqrt{(a1-b1)^(2)+(a2-b2)^(2)}` or

`d(B,A)=\sqrt{(b1-a1)^(2)+(b2-a2)^(2)}`

Now for
`A=(-2,-4)` and
`B=(-8,4)`

The distance is

`d(A,B)=\sqrt{(-2-(-8))^(2)+(-4-4)^(2)}=\sqrt{6^(2)+(-8)^(2)}=√(36+64)=√(100)=10`

The equation I used derives from operator norm.

Given a point
`C=(c1,c2)` in the plane, the distance between C and (0,0) is

`||C||=\sqrt{(c1)^(2)+(c2)^(2)}`

`||.||` is the operator norm.

To find the distance between A and B we apply
`||.||` to the diference
`(A-B)` or
`(B-A)` in order to obtain the distance equation
`d(A,B)` or
`d(B,A)`

Notice that
`d(A,B)=d(B,A)`

Answer 2

Imagine it as a right angled triangle, and you can use Pythagoras' theorem to work it out.

Difference of x values (length of x) = -8 - -2 = -6
Different of y values (length of y) = 4 - - 4 = 8

AB =
`\sqrt{ -6^(2) + 8^(2) } = 10`