Question
Two particles travel along the space curves $\mathbf{r}(t)$ and $\mathbf{u}(t)$.If the particles collide, do their paths $\mathbf{r}(t)$ and $\mathbf{u}(t)$ intersect?
Step 1
These are parametric equations that describe the paths of the two particles in space as a function of time. Show more…
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Two particles travel along the space curves $\mathbf{r}(t)$ and $\mathbf{u}(t)$. If $\mathbf{r}(t)$ and $\mathbf{u}(t)$ intersect, will the particles collide?
Vector-Valued Functions
Two particles travel along the space curves $r(t)$ and $u(t) .$ A collision will occur at the point of intersection $P$ when both particles are at $P$ at the same time. Do the particles collide? Do their paths intersect? $$\begin{aligned}&\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}\\&\mathbf{u}(t)=(-2 t+3) \mathbf{i}+8 t \mathbf{j}+(12 t+2) \mathbf{k}\end{aligned}$$
Two particles travel along the space curves $r(t)$ and $u(t) .$ A collision will occur at the point of intersection $P$ when both particles are at $P$ at the same time. Do the particles collide? Do their paths intersect? $$\begin{array}{l}\mathbf{r}(t)=t^{2} \mathbf{i}+(9 t-20) \mathbf{j}+t^{2} \mathbf{k} \\ \mathbf{u}(t)=(3 t+4) \mathbf{i}+t^{2} \mathbf{j}+(5 t-4) \mathbf{k}\end{array}$$
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