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

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

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 $x = 3t - 1, \quad y = 2t + 4, \quad z = 5 - t$ Determine whether the point ever reaches the position $(8, -2, 1)$. If so, find the value of $t$.

Does the line

pass through the point $(17, 14, 5)$? If so, find the value of $t$.

Determine whether the trajectory defined by

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 $\mathbf{r}(t) = \bigl(t - \sin t, 1 - \cos t\bigr)$ Does it pass through the point $(\pi, 2)$? If so, find the value of $t$.

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

Determine whether the trajectories $\mathbf{r}_1(t) = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} + t \begin{pmatrix} 2 \\ 3 \\ 4 \end{pmatrix} \quad \text{and} \quad \mathbf{r}_2(s) = \begin{pmatrix} -1 \\ 4 \\ 2 \end{pmatrix} + s \begin{pmatrix} 3 \\ -1 \\ 2 \end{pmatrix}$ 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 $\mathbf{r} = \vec{OA} + t\vec{AB}$, 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.