91Ó°ÊÓ

Find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1}\). $$ \mathbf{r}(t)=(t+1) \mathbf{i}+3 t \mathbf{j}+t^{2} \mathbf{k} ; t_{1}=1 $$

Give a proof of the indicated property for two-dimensional vectors. Use \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle, \mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle,\) and \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). \(\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}\)

The curve defined by \(x=a \cos t, y=a \sin t, z=c t\) is a helix. Hold \(a\) fixed and use a CAS to obtain a parmetric plot of the helix for various values of \(c\). What effect does \(c\) have on the curve?

Find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1}\). $$ x=t, y=t^{2}, z=t^{3} ; t_{1}=2 $$

Give a proof of the indicated property for two-dimensional vectors. Use \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle, \mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle,\) and \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). \(0 \cdot \mathbf{u}=0\)

Find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1}\). $$ x=e^{-t}, y=2 t, z=e^{t} ; t_{1}=0 $$

Find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1}\). $$ \mathbf{r}(t)=(t-2)^{2} \mathbf{i}-t^{2} \mathbf{j}+t \mathbf{k} ; t_{1}=2 $$

Give a proof of the indicated property for two-dimensional vectors. Use \(\mathbf{u}=\left\langle u_{1}, u_{2}\right\rangle, \mathbf{v}=\left\langle v_{1}, v_{2}\right\rangle,\) and \(\mathbf{w}=\left\langle w_{1}, w_{2}\right\rangle\). \(\mathbf{u} \cdot \mathbf{u}=\|\mathbf{u}\|^{2}\)

Given the two nonparallel vectors \(\mathbf{a}=3 \mathbf{i}-2 \mathbf{j}\) and \(\mathbf{b}=-3 \mathbf{i}+4 \mathbf{j}\) and another vector \(\mathbf{r}=7 \mathbf{i}-8 \mathbf{j},\) find scalars \(k\) and \(m\) such that \(\mathbf{r}=k \mathbf{a}+m \mathbf{b}\)

Find the tangential and normal components \(\left(a_{T}\right.\) and \(\left.a_{N}\right)\) of the acceleration vector at \(t .\) Then evaluate at \(t=t_{1}\). $$ \mathbf{r}(t)=\left(t-\frac{1}{3} t^{3}\right) \mathbf{i}-\left(t+\frac{1}{3} t^{3}\right) \mathbf{j}+t \mathbf{k} ; t_{1}=3 $$