91Ó°ÊÓ

Displacement in the first putt,${}^{\stackrel{\mathbf{→}}{\mathbf{a}}\mathbf{}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{66}\mathbf{}\mathbf{m}\mathbf{}\mathbf{north}}$

Displacement in the second putt,${}^{\stackrel{\mathbf{→}}{\mathbf{b}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{83}\mathbf{}\mathbf{m}\mathbf{}\mathbf{southeast}}$

Displacement in the third putt,${}^{\stackrel{\mathbf{→}}{\mathbf{c}}\mathbf{}\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{91}\mathbf{}\mathbf{m}\mathbf{}\mathbf{southwest}}\mathbf{}$

The vectors can be added geometrically by placing them head to tail. The vector that connects the tail of first to head of the last vector is the vector sum.

The displacement vector ${}^{\stackrel{\mathbf{→}}{\mathbf{R}}}$can be written as,

${}^{\stackrel{\mathbf{→}}{\mathbf{R}}\mathbf{=}\stackrel{\mathbf{→}}{\mathbf{a}}\mathbf{+}\stackrel{\mathbf{→}}{\mathbf{b}}\mathbf{+}\stackrel{\mathbf{→}}{\mathbf{c}}}$ … (i)

Consider the directions as East is along x axis and North is along y axis. Thus the given vectors ${}^{\stackrel{\mathbf{→}}{\mathbf{a}}}$, ${}^{\stackrel{\mathbf{→}}{\mathbf{b}}}$and ${}^{\stackrel{\mathbf{→}}{\mathbf{c}}}$can be drawn as:

The given vectors can be written in terms of their components as,

$\stackrel{\mathbf{→}}{\mathbf{a}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{66}\mathbf{}\hat{\mathbf{j}}$

$\stackrel{\mathbf{→}}{\mathbf{b}}\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{83}\mathbf{cos}\left(-{45}^{°}\right)\widehat{\mathbf{}\mathbf{i}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{83}\mathbf{sin}\left(-{45}^{°}\right)\mathbf{}\hat{\mathbf{j}}$

$\stackrel{\mathbf{→}}{\mathbf{c}}\mathbf{}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{91}\mathbf{cos}\left(-{135}^{°}\right)\mathbf{}\hat{\mathbf{i}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{9}\mathbf{sin}\left(-{135}^{°}\right)\mathbf{}\hat{\mathbf{j}}$

Using equation (i), the resultant vector can be obtained as,

$\stackrel{\mathbf{→}}{\mathbf{r}}\mathbf{}\mathbf{=}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{a}}\mathbf{+}\stackrel{\mathbf{→}}{\mathbf{b}}\mathbf{+}\mathbf{}\stackrel{\mathbf{→}}{\mathbf{c}}$

$\mathbf{=}\mathbf{}\mathbf{0}\mathbf{.}\mathbf{64}\mathbf{}\hat{\mathbf{i}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{72}\mathbf{}\hat{\mathbf{j}}\mathbf{}$

The magnitude of the resultant vector is,

$$\begin{gathered} \left|\stackrel{→}{r}\right|\mathbf{}\mathbf{=}\sqrt{\mathbf{0}\mathbf{.}{\mathbf{65}}^{\mathbf{2}}\mathbf{+}\mathbf{}\mathbf{1}\mathbf{.}{\mathbf{72}}^{\mathbf{2}}} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\mathbf{.}\mathbf{84}\mathbf{}\mathit{m} \end{gathered}$$

Thus, the magnitude of the displacement needed to get the ball into the hole on the first putt is 1.84 m.

The direction of the displacement vector is calculated as:

$$\begin{gathered} \mathit{θ}\mathbf{=}\mathbf{}\mathit{t}\mathit{a}{\mathit{n}}^{\mathbf{-}\mathbf{1}}\mathbf{}\mathbf{\left(}\frac{\mathbf{1}\mathbf{.}\mathbf{72}}{\mathbf{0}\mathbf{.}\mathbf{65}}\mathbf{\right)} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}\mathbf{}\mathbf{69}\mathbf{.}{\mathbf{3}}^{\mathbf{°}} \end{gathered}$$

Thus the direction of the displacement needed to get the ball into the hole on the first putt is ${}^{{\mathbf{63}}^{\mathbf{°}}\mathbf{}\mathbf{north}\mathbf{}\mathbf{of}\mathbf{}\mathbf{east}}$.