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**heathrowjohnny** $\displaystyle \bold{r}(t) = t^{2} \bold{i} - 4t \bold{j} -t^{2} \bold{k} $

$\displaystyle \bold{v}(t) = 2t \bold{i} -4 \bold{j} -2t \bold{k} $

$\displaystyle r(t) = \sqrt{2t^{4} + 16t^{2}} $

$\displaystyle v(t) = \sqrt{8t^{2} + 16} $.

Find the cosine of the angle between $\displaystyle \bold{r} $ and $\displaystyle \bold{v} $. For what values of $\displaystyle t $ is $\displaystyle \bold{r} $ perpendicular to $\displaystyle \bold{v} $? Parallel?

So $\displaystyle \cos \theta = \frac{ \bold{r}(t) \cdot \bold{v}(t)}{r(t) \cdot v(t)} $

$\displaystyle \cos \theta = \frac{4t^{3} + 16t}{2t \sqrt{4t^{4} + 40t^{2} + 64}} $

So $\displaystyle t = 0 $ or $\displaystyle t = \pm 2i $ for $\displaystyle \bold{r} \perp \bold{v} $.

But for $\displaystyle \bold{r} \parallel \bold{v} $ I get $\displaystyle t = 0 $. But this is undefined. So there are no values of $\displaystyle t $ for which $\displaystyle \bold{r} $ and $\displaystyle \bold{v} $ are parallel?