Need Help Understanding Why $Delta vec{v}$ is Perpendicular to $vec{v}$

Question

I am confused about the statement of how vector $Delta vec{v}$ is perpendicular to vector. I highlighted the statement in pink. I ended up copying the image of the right vector $vec{v}$ in the velocity isosceles triangle and moved its tail to touch the tail of vector $vec{v}(t)$. It does not look perpendicular so could someone clarify my misunderstanding?

mike stone · Accepted Answer

This is only true if ${bf v}$ just changes direction but keeps the same length. If ${bf v}$ and ${bf v}+ Delta {bf v}$ have the same length, then
$$
{bf v}cdot {bf v}= ({bf v}+ Delta {bf v})cdot  ({bf v}+ Delta {bf v})= {bf v}cdot {bf v}+ 2{bf v}cdot Delta{bf v}+ (Delta {bf v})cdot (Delta {bf v})
$$
so as $Delta {bf v}$ gets small we must have ${bf v}cdot Delta{bf v}=0$. i.e. they become perpendicular.

Need Help Understanding Why $Delta vec{v}$ is Perpendicular to $vec{v}$

One Answer

Add your own answers!

Ask a Question