MHB How do I find the unit tangent to the trajectory as a function of time?

[Kris1]

A particle moves such that its position vector, as a function of time is
r(t) = (5*t^2)*i + (3*t^2)*j − (5*t)*k

Im trying to find the unit tangent to the trajectory as a function of time. However I can't seem to find any formula of how to do this.

Can someone please help me with a formula or as to how to go about this?

 

[Chris L T521]

A particle moves such that its position vector, as a function of time is
r(t) = (5*t^2)*i + (3*t^2)*j − (5*t)*k

Im trying to find the unit tangent to the trajectory as a function of time. However I can't seem to find any formula of how to do this.

Can someone please help me with a formula or as to how to go about this?

The unit tangent vector is defined to be
\[\mathbf{T}(t)=\frac{\mathbf{r}^{\prime}(t)} {|\mathbf{r}^{\prime}(t)|}.\]
Since you're given $\mathbf{r}(t)$, it shouldn't be too difficult computing $\mathbf{r}^{\prime}(t)$ and it's norm $|\mathbf{r}^{\prime}(t)|$.

Can you take things from here?

 

[Kris1]

Thanks for the help

The answer I worked out to be was
(10*t*i+6*t*j-5*k)/(sqrt(136*t^2+25))

I at first got it wrong because I tried this answer
(10*t+6*t-5)/(sqrt(136*t^2+25))

But that didn't account for the i,j,k in the original vector hence the answer is
(10*t*i+6*t*j-5*k)/(sqrt(136*t^2+25))

Thanks for the help guys and I hope this can help someone else