Question

is the following function a proper distance function? why? explain your answer. measure the distance between (0, 0, 0) and (0, 1, 0)

Answer 1

Using distance formula,

The given distance function is not proper distance function. the distance between (0,0,0) and (0,1,0) is 1 unit .

Proper distance function: A function is a proper distance function if the following conditions are satisfied

• d(x, y) > 0 for every x and y and
• d(x, y)=0 if and only if x=y
• d(x, y)=d(y,x)

we have given distance function is

d(x,y) = ∑(xᵢ - yᵢ)³

we want to check that the given function is proper distance function or not.

for this we check the above three conditions

a) we see that d(x,y) is may be positive due to odd powers of difference of x and y values .

that is d(x,y) may be greater than zero.

so, it is not necessary to check the other conditions .

condition of proper distance function is not satisfied by given distance function so, it is not proper distance function.

Now, we shall calculate the distance between (0,0,0) and (0,1,0).

distance formula for three cordinate is

d(x,y,z) = √(x₂-x₁)²+(y₂-y₁)²+(z₂-z₁)²

x₁= y₁= z₁= 0, x₂= 0 , y₂= 1 , z₂= 0

putting all the values in above distance formula we get, d(x,y) = √(0) + (1-0)²+0 = 1

Hence, given distance function is not proper distance function and the distance between (0, 0, 0) and (0, 1, 0) is 1 units .

To learn more about Distance, refer :

https://brainly.com/question/17273444

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complete question

d(x,y) = ∑(xᵢ - yᵢ)³

is the above function a proper distance function? why? explain your answer. measure the distance between (0, 0, 0) and (0, 1, 0)