Question

Show that the condition for the vector w to be equidistant from u and v is equivalent to: u²−2wu=v²−2wv

Answer 1

To show that the vector w is equidistant from vectors u and v, we need to prove that u² - 2wu = v² - 2wv. We can start by finding the distance between w and u using the formula for the dot product. Expanding and simplifying the expression, we can conclude that if w is equidistant from u and v, then u² - 2wu = v² - 2wv.

Step-by-step explanation:

To show that the vector w is equidistant from vectors u and v, we need to prove that u² - 2wu = v² - 2wv.

We can start by finding the distance between w and u:

Distance between w and u = ||w - u||2

Using the formula for the dot product, we can rewrite this as:

Distance between w and u = (w - u) · (w - u)

Expanding and simplifying, we get:

Distance between w and u = w² - 2wu + u²

Notice that this is the same as the expression we want to prove, so we can conclude that if w is equidistant from u and v, then u² - 2wu = v² - 2wv.