Question Type 1: Determining speed of an object for a given trajectory Exercises

An object moves with position vector $\mathbf{r}(t) = \begin{pmatrix} 4 \\ -2 \\ 1 \end{pmatrix} + t \begin{pmatrix} -3 \\ 6 \\ 2 \end{pmatrix}$. Calculate its speed.

An object follows the path $\mathbf{r}(t) = (2, 1, -1) + t(3, 4, 5)$. Determine whether it passes through the point $(11, 13, 19)$, and find $t$ if it does.

Determine whether the line $\mathbf{r}(t) = (5, 5, 5) + t(2, -3, 4)$ passes through the point $(17, -1, 21)$ and find the corresponding value of $t$ if it does.

An object moves with a position vector given by $\mathbf{r}(t) = \begin{pmatrix} 3 \\ -2 \\ 5 \end{pmatrix} + t \begin{pmatrix} -2 \\ 3 \\ 1 \end{pmatrix}$.

Determine whether the object reaches the point $(-1, 7, 8)$, and find the value of $t$ if it does.

An object travels on the path $\mathbf{r}(t) = (0,0,0) + t(2,2,1)$, where $t$ is time. Find the speed of the object.

Find the speed of an object moving along $\mathbf{r}(t) = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} + t \begin{pmatrix} 0 \\ 5 \\ -12 \end{pmatrix}$.

Does the line $\boldsymbol{r}(t) = (0, 0, 0) + t(1, 2, 4)$ pass through the point $(7, 14, 28)$? If so, find the value of $t$.

An object moves with constant velocity such that its position vector at time $t$ is given by $\mathbf{r}(t) = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix} + t \begin{pmatrix} 8 \\ 7 \\ 2 \end{pmatrix}$.

Find the speed of the object.

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

Given $\mathbf{r}(t) = \begin{pmatrix} -3 \\ 5 \\ 0 \end{pmatrix} + t \begin{pmatrix} 6 \\ -2 \\ 1 \end{pmatrix}$, determine the speed of the object.

An object moves along the path $\mathbf{r}(t) = (2,-1,3) + t(4,0,-3)$. Calculate its constant speed.

For an object with position vector $\mathbf{r}(t) = \begin{pmatrix} -1 \\ 2 \\ 4 \end{pmatrix} + t \begin{pmatrix} 5 \\ -1 \\ 3 \end{pmatrix}$, determine if the object ever reaches the point $(9, -3, 13)$. Find the value of $t$ if it does, or explain why not if it does not.