Calculus with Vector Functions — Question 2

Question 2

Let $\vec u(t)=\left\langle t,t^2,1\right\rangle$ and $\vec v(t)=\left\langle e^t,\cos t,t\right\rangle$. Compute $\frac d{dt}(\vec u\cdot\vec v)$ and $\frac d{dt}(\vec u\times\vec v)$ at $t=0$ using product rules, then verify the dot-product result by expanding first.

Question 2 – Solution

Strategy Use $(\vec u\cdot\vec v)'=\vec u'\cdot\vec v+\vec u\cdot\vec v'$ and $(\vec u\times\vec v)'=\vec u'\times\vec v+\vec u\times\vec v'$.

Values at zero $\vec u(0)=\left\langle 0,0,1\right\rangle$, $\vec u'(0)=\left\langle 1,0,0\right\rangle$, $\vec v(0)=\left\langle 1,1,0\right\rangle$, and $\vec v'(0)=\left\langle 1,0,1\right\rangle$.

Results Thus $$(\vec u\cdot\vec v)'(0)=1+1=\boxed 2,$$ and $$(\vec u\times\vec v)'(0)=\left\langle 0,0,1\right\rangle+\left\langle 0,1,0\right\rangle=\boxed{\left\langle 0,1,1\right\rangle}.$$ Expanding $\vec u\cdot\vec v=te^t+t^2\cos t+t$ and differentiating also gives 2 at zero.