Vector Proof Question

Question

Suppose v and w are vectors in Rⁿ . Show that || v − w || ≥ || v || − || w ||

Answer 1

Using the triangular inequality, we know that for any two vectors a and b, we have

|| a + b || ≤ ||a|| + ||b||

Now notice that || v || = || v - w + w ||, where we just add and subtract the term w then using triangular inequality we get

|| v || ≤ || v - w || + || w ||

Now subtract || w || on both sides, we get

|| v || - || w || ≤ || v - w ||

Answer 2

Claim: || v − w || ≥ || v || − || w || holds for all vectors v & w in R^n.

Proof:

Start with the triangle inequality, || x + y || ≤ || x || + || y ||, which is true for all vectors x and y in R^n.

Now substitute v - w for x and w for y, which gives the following:

||(v - w) + w || ≤ || (v - w) || + || w ||, and simplifies to: || v || ≤ || (v - w) || + || w ||.

Finally, subtract || w || from both sides to get, || v || - || w || ≤ || (v - w) ||, which is the desired inequality (sides reversed BUT preserving the direction of inequality) Q.E.D.