IB Mathematics Applications & Interpretation Question Type 2: Determining with an Object Given a Trajectory Passes a Specific Point Exercises

A particle's position at time $t$ is given by the parametric equations

$$x=1+4t,y=2-2t,z=3+t$$

Determine whether there is a value of $t$ for which the particle occupies the point $(5,-2,5)$.

A point moves according to the parametric equations $$x=3t-1,y=2t+4,z=5-t$$ Determine whether the point ever reaches the position $(8,-2,1)$. If so, find the value of $t$.

Does the line $$\mathbf{r}(t)=(1,0,1)+t(8,7,2)$$ pass through the point $(17,14,5)$? If so, find the value of $t$.

Determine whether the trajectory defined by $$\mathbf{r}(t)=\left(\begin{array}{c}2\\ 3\end{array}\right)+t\left(\begin{array}{c}5\\ -2\end{array}\right)$$ passes through the point $(17,-1)$. If it does, find the corresponding parameter $t$.

Does the parameterised curve $$\mathbf{r}(t)=\left(\sqrt{2}\cos t,\sqrt{2}\sin t\right)$$ pass through $(1,1)$? If so, find $t$ in the interval $[0,2\pi )$.

A cycloid is defined by the vector equation $$\mathbf{r}(t)=(t-\sin t,1-\cos t)$$ Does it pass through the point $(\pi ,2)$? If so, find the value of $t$.

Does the circle parameterised by $$\mathbf{r}(t)=(3\cos t,3\sin t)$$ pass through $(-\frac{3}{2},\frac{3\sqrt{3}}{2})$? Find $t$.

Determine whether the trajectories $${\mathbf{r}}_1(t)=\left(\begin{array}{c}1\\ 0\\ 0\end{array}\right)+t\left(\begin{array}{c}2\\ 3\\ 4\end{array}\right)\text{and}{\mathbf{r}}_2(s)=\left(\begin{array}{c}-1\\ 4\\ 2\end{array}\right)+s\left(\begin{array}{c}3\\ -1\\ 2\end{array}\right)$$ intersect. If so, find the values of $t$ and $s$ and the point of intersection.

The line $L$ passes through the points $A(1,2,3)$ and $B(4,5,6)$. The vector equation of $L$ is given by $\mathbf{r}=\overrightarrow{OA}+t\overrightarrow{AB}$, where $t$ is a scalar parameter.

Find the value of $t$ for which the point $C(7,8,9)$ lies on $L$.

A projectile’s position is given by $$\mathbf{r}(t)=(0,0,0)+t(5,10,0)+t^2(0,-5,0)$$ Does it pass through $(10,5,0)$? If yes, find $t$.

Does the helix $$\mathbf{r}(t)=(\cos t,\sin t,t)$$ pass through the point $\left(\frac{1}{2},\frac{\sqrt{3}}{2},\frac{\pi }{3}\right)$? Find $t$ if it does.