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The displacements vector are

$$\begin{gathered} \stackrel{鈬赌}{{\mathrm{d}}_1}=\left(8\mathrm{m}\right)\hat{\mathrm{i}}+\left(6\mathrm{m}\right)\hat{\mathrm{j}} \\ \stackrel{鈬赌}{{\mathrm{d}}_1}=\left(6\mathrm{m}\right)\hat{\mathrm{j}} \\ \stackrel{鈬赌}{{\mathrm{d}}_3}=\left(8\mathrm{m}\right)\hat{\mathrm{i}} \end{gathered}$$

The pit is at coordinates (8 m, 6 m).

If the two vectors are directed exactly in the same direction, then they can be added using the simple addition rules. But if their directions are different, then we have to use the components of vectors along unit vectors and add them. Using the vector addition laws, we can add any two vectors.

Formula:

$\stackrel{鈬赌}{a}+\stackrel{鈬赌}{b}=\stackrel{鈬赌}{c}$

The ball is at origin $\left(0,0\right)$. We have to put ball in the hole at $\left(8,12\right)$. We can use displacements given as ,$\stackrel{鈬赌}{d_1},\stackrel{鈬赌}{d_2}\mathrm{and}\stackrel{鈬赌}{{\mathrm{d}}_3}$. The ball should not go inside the pit at ( 8, 6 ). The displacements of ball should be multiples of $\stackrel{鈬赌}{d_1},\stackrel{鈬赌}{d_2}\mathrm{and}\stackrel{鈬赌}{{\mathrm{d}}_3}$.

First, the ball goes along $\stackrel{鈬赌}{d_2}$and then $\stackrel{鈬赌}{d_1}$, by using vector addition law.

$\stackrel{鈬赌}{d}=\stackrel{鈬赌}{d_2}+\stackrel{鈬赌}{{\mathrm{d}}_1}$

Substitute the given values of the vectors.

$$\begin{gathered} \stackrel{鈬赌}{\mathrm{d}}=\left(6\mathrm{m}\right)\hat{\mathrm{j}}+\left(8\mathrm{m}\right)\hat{\mathrm{i}}+\left(6\mathrm{m}\right)\hat{\mathrm{j}} \\ \stackrel{鈬赌}{\mathrm{d}}=\left(8\mathrm{m}\right)\hat{\mathrm{i}}+\left(12\mathrm{m}\right)\hat{\mathrm{j}} \end{gathered}$$

By using this sequence, the ball can be put in the hole at (8, 12 ) .

Another way the ball goes is along , ${\stackrel{鈬赌}{d}}_2,{\stackrel{鈬赌}{d}}_2\mathrm{and}\mathrm{then}{\stackrel{鈬赌}{\mathrm{d}}}_3$.

According to the vector addition law,

$\stackrel{鈬赌}{d}={\stackrel{鈬赌}{d}}_2+{\stackrel{鈬赌}{d}}_2+{\stackrel{鈬赌}{d}}_3$

Substitute the given values of the vectors.

$$\begin{gathered} \stackrel{鈬赌}{\mathrm{d}}=\left(6\mathrm{m}\right)\hat{\mathrm{j}}+\left(6\mathrm{m}\right)\hat{\mathrm{j}}+\left(8\mathrm{m}\right)\hat{\mathrm{i}} \\ \stackrel{鈬赌}{\mathrm{d}}=\left(8\mathrm{m}\right)\hat{\mathrm{i}}+\left(6\mathrm{m}\right)\hat{\mathrm{j}} \end{gathered}$$

By using this sequence, ball can be put into the hole at (8,12 ).

Thus, the sequence will be${\stackrel{鈬赌}{d}}_2,{\stackrel{鈬赌}{d}}_1\mathrm{or}{\stackrel{鈬赌}{\mathrm{d}}}_2,{\stackrel{鈬赌}{\mathrm{d}}}_2,{\stackrel{鈬赌}{\mathrm{d}}}_3$