IB Mathematics Applications & Interpretation Question Type 1: Determining Speed of an Object for a Given Trajectory Exercises

An object moves with position vector $\mathbf{r}(t)=\left(\begin{array}{c}4\\ -2\\ 1\end{array}\right)+t\left(\begin{array}{c}-3\\ 6\\ 2\end{array}\right)$. 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 $\mathbf{r}(t)=\left(\begin{array}{c}3\\ -2\\ 5\end{array}\right)+t\left(\begin{array}{c}-2\\ 3\\ 1\end{array}\right)$.

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 $\mathbf{r}(t)=\left(\begin{array}{c}1\\ 2\\ 3\end{array}\right)+t\left(\begin{array}{c}0\\ 5\\ -12\end{array}\right)$.

Does the line $\mathbf{r}(t)=(0,0,0)+t(1,2,4)$ 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 $\mathbf{r}(t)=\left(\begin{array}{c}1\\ 0\\ 1\end{array}\right)+t\left(\begin{array}{c}8\\ 7\\ 2\end{array}\right)$.

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 $\mathbf{r}(t)=\left(\begin{array}{c}-3\\ 5\\ 0\end{array}\right)+t\left(\begin{array}{c}6\\ -2\\ 1\end{array}\right)$, 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 $\mathbf{r}(t)=\left(\begin{array}{c}-1\\ 2\\ 4\end{array}\right)+t\left(\begin{array}{c}5\\ -1\\ 3\end{array}\right)$, 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.