Question

In a game held within a three dimensional maze, you must move your game piece from start, at xyz coordinates $\mathbf{(}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{,}\mathbf{0}\mathbf{)}$, to finish, at coordinates $\mathbf{(}\mathbf{-}\mathbf{2}\mathbf{}\mathbf{cm}\mathbf{,}\mathbf{4}\mathbf{}\mathbf{cm}\mathbf{,}\mathbf{-}\mathbf{4}\mathbf{}\mathbf{cm}\mathbf{)}\mathbf{.}$The game piece can undergo only the displacements (in centimeters) given below. If, along the way, the game piece lands at coordinates$\mathbf{(}\mathbf{-}\mathbf{5}\mathbf{}\mathbf{cm}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{}\mathbf{cm}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{}\mathbf{cm}\mathbf{)}$or $\mathbf{(}\mathbf{5}\mathbf{}\mathbf{cm}\mathbf{,}\mathbf{2}\mathbf{}\mathbf{cm}\mathbf{,}\mathbf{-}\mathbf{1}\mathbf{}\mathbf{cm}\mathbf{)}$, you lose the game. Which displacements and in what sequence will get your game piece to finish?

role="math" localid="1657006865865" $$\begin{gathered} \mathit{p}\mathbf{}\mathbf{⃗}\mathbf{=}\mathbf{-}\mathbf{7}\hat{\mathbf{i}}\mathbf{+}\mathbf{2}\hat{\mathbf{j}}\mathbf{-}\mathbf{3}\hat{\mathbf{k}}\mathbf{} \\ \mathit{r}\mathbf{}\mathbf{⃗}\mathbf{=}\mathbf{2}\hat{\mathbf{i}}\mathbf{-}\mathbf{3}\hat{\mathbf{j}}\mathbf{+}\mathbf{2}\hat{\mathbf{k}} \\ \mathit{q}\mathbf{}\mathbf{⃗}\mathbf{=}\mathbf{2}\hat{\mathbf{i}}\mathbf{-}\hat{\mathbf{j}}\mathbf{+}\mathbf{4}\hat{\mathbf{k}}\mathbf{} \\ \mathit{s}\mathbf{}\mathbf{⃗}\mathbf{=}\mathbf{3}\hat{\mathbf{i}}\mathbf{+}\mathbf{5}\hat{\mathbf{j}}\mathbf{-}\mathbf{3}\hat{\mathbf{k}} \end{gathered}$$

Short answer

The sequence of the displacements will be$\mathrm{p}⃗-\mathrm{s}⃗-\mathrm{r}⃗.$

Step-by-step solution

Given information

• Starting coordinates$(0,0,0)$
• Finishing coordinates $(-2\mathrm{cm},4\mathrm{cm},-4\mathrm{cm})$
• Coordinates that are not allowed$(-5\mathrm{cm},-1\hspace{0.33em}\mathrm{cm},-1\hspace{0.33em}\mathrm{cm})\mathrm{and}(5\mathrm{cm},2\mathrm{cm},-1\mathrm{cm})$

To understand the concept

The vector quantity has magnitude as well as direction. The vectors cannot be added using simple mathematical operations. Two vectors can be added only when they are of the same type and nature. To add the vectors, we can use the laws of vector addition.

Formula:

$\overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{B}}=\overrightarrow{\mathrm{C}}$

To find the displacements in sequence to finish the game piece

Let us start with displacement $p⃗=-7\hat{i}+2\hat{j}-3\hat{k}$

After that, we can use displacement$s⃗=3\hat{i}+5\hat{j}-3\hat{k}$

By adding these two displacements we can reach a point say A.

role="math" localid="1657007161563" $$\begin{gathered} p⃗+s⃗=(-7\hat{i}+2\hat{j}-3\hat{k})+(3\hat{i}+5\hat{j}-3\hat{k}) \\ p⃗+s⃗=(-4\hat{i}+7\hat{j}-"6\hat{k}) \end{gathered}$$

Now, we will use displacement

$$\begin{gathered} \mathrm{r}⃗=(2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+2\hat{\mathrm{k}}) \\ \overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{r}}=(-4\hat{\mathrm{i}}+7\hat{\mathrm{j}}-6\hat{\mathrm{k}})+(2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+2\hat{\mathrm{k}}) \\ \overrightarrow{\mathrm{A}}+\overrightarrow{\mathrm{r}}=(-2\hat{\mathrm{i}}+4\hat{\mathrm{j}}-4\hat{\mathrm{k}}) \end{gathered}$$

We can ensure to avoid the points$(-5\mathrm{cm},-1\hspace{0.33em}\mathrm{cm},-1\hspace{0.33em}\mathrm{cm})$ as we have not moved to negative y axis.

And also, it is ensured that we cannot come across point$(5\mathrm{cm},2\mathrm{cm},-1\mathrm{cm})$as we have not moved to positive x axis.

Thus, the sequence of the displacements will be $\overrightarrow{p}-\overrightarrow{s}-\overrightarrow{r}$.