I need help with vector derivation!

[magdaddy101]

This is the problem: Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement.
If r(t) is a differentiable vector function, then d/dt |r(t)| = |r'(t)|.


any help would be much appreciated. thanks in advance!

 

[galactus]

Just to confirm. What do the vertical bars represent?. Generally, they represent absolute value.

Do you perhaps mean the norm, ||r(t)||?.

We can prove that \(\displaystyle \frac{d}{dt}[||r(t)||]\neq||r'(t)||\) easy enough by using an example.

Let \(\displaystyle r(t)=cos(t)i+sin(t)j+k\)

\(\displaystyle ||r(t)||=\sqrt{\underbrace{cos^{2}(t)+sin^{2}(t)}_{\text{equal to 1}}+1^{2}}=\sqrt{2}\)

\(\displaystyle \frac{d}{dt}[||r(t)||]=0\)

\(\displaystyle r'(t)=-sin(t)i+cos(t)j\)

\(\displaystyle ||r'(t)||=\sqrt{(-sin(t))^{2}+(cos(t))^{2}}=1\)

\(\displaystyle 0\neq 1\) and \(\displaystyle \frac{d}{dt}[||r(t)||]\neq||r'(t)||\)

 

[magdaddy101]

thanks for the reply, but the vertical bars do mean absolute value. So in this case, it's referring to the vector lengths, I guess.