In each case, find the x- and y-components of vector$\overrightarrow{\mathbf{A}}$: (a)$\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{-}\mathbf{6}\mathbf{.}\mathbf{3}\hat{\mathbf{j}}$; (b)$\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{11}\mathbf{.}\mathbf{2}\hat{\mathbf{j}}\mathbf{-}\mathbf{9}\mathbf{.}\mathbf{91}\hat{\mathbf{i}}$; (c)$\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{-}\mathbf{15}\mathbf{.}\mathbf{0}\hat{\mathbf{i}}\mathbf{+}\mathbf{22}\mathbf{.}\mathbf{4}\hat{\mathbf{j}}$; (d)$\overrightarrow{\mathbf{A}}\mathbf{=}\mathbf{5}\mathbf{.}\mathbf{0}\overrightarrow{\mathbf{B}}$, where$\overrightarrow{\mathbf{B}}\mathbf{=}\mathbf{4}\hat{\mathbf{i}}\mathbf{-}\mathbf{6}\hat{\mathbf{j}}$.

$•Thegivenvectorsare,\overrightarrow{A}=5.0\widehat{i}-6.3\widehat{j},\overrightarrow{A}=11.2\widehat{j}-9.91\widehat{i},\overrightarrow{A}=-15.0\widehat{i}+22.4\widehat{j}and\overrightarrow{B}=4\widehat{i}-6\widehat{j}$

$•Therelationbetweenvector\overrightarrow{A}andvector\overrightarrow{B}isgivenby\overrightarrow{A}=5.0\overrightarrow{B}$

For a given vector$\overrightarrow{A}=A_x\widehat{i}+A_y\widehat{j}$, A_x is the component in x- direction and A_y is the component in y-direction and are the unit vectors in x and y directions respectively

Part(a)

The given vector $\stackrel{\rightharpoonup }{A}$ can be represented as,

$\overrightarrow{A}=5.0\widehat{i}-6.3\widehat{j}$

The given vector can be represented in the form of $\overrightarrow{A}=A_x\widehat{i}+A_y\widehat{j}$ as,

$\overrightarrow{A}=5.0\widehat{i}+\left(-6.3\right)\widehat{j}$

The components of the given vector A_x and A_y will be,

A_x = 5.0

A_y = -6.3

Thus, the component of the given vector in the x-direction is A_x = 5.0 and the component in the y-direction is A_y = -6.3 .

Part(b)

The given vector$\overrightarrow{A}$ can be represented as,

$\overrightarrow{A}=11.2\widehat{j}-9.91\widehat{i}$

The given vector can be represented in the form of $\overrightarrow{A}=A_x\widehat{i}+A_y\widehat{j}$ as,

$\overrightarrow{A}=\left(-9.91\right)\widehat{i}+11.2\widehat{j}$

The components of the given vector A_x and A_y can be represented as,

$\begin{array}{l}{A}_x=-9.11\\ A_y=11.2\end{array}$

Thus, the component of the given vector in the x-direction is A_X = -9.11 and the component in the y-direction is A_y= 11.2

Part(c)

The given vector$\overrightarrow{A}$ can be represented as,

$\overrightarrow{A}=-15.0\widehat{i}+22.4\widehat{j}$

The given vector can be represented in the form of $\overrightarrow{A}=A_x\widehat{i}+A_y\widehat{j}$ as,

$\overrightarrow{A}=\left(-15.0\right)\widehat{i}+22.4\widehat{j}$

The components of the given vector A_x and A_y can be represented as,

$\begin{array}{l}{A}_x=-15.0\\ A_y=22.4\end{array}$

Thus, the component of the given vector in the x-direction is A_x= -15.0 and the component in the y-direction is A_y= 22.4 .

Part(d)

The given vector $\overrightarrow{B}$ can be represented as,

$\overrightarrow{B}=4\widehat{i}-6\widehat{j}$

The relation between vector $\overrightarrow{A}$and vector $\overrightarrow{B}$can be expressed as,

$\overrightarrow{A}=5.0\overrightarrow{B}$

$Substitute4\widehat{i}-6\widehat{j}for\overrightarrow{B}$

$\begin{array}{c}\overrightarrow{A}=5.0\overrightarrow{B}\\ =5.0\left(4\widehat{i}-6\widehat{j}\right)\\ =\left(5.0\times 4\right)\widehat{i}-\left(5.0\times 6\right)\widehat{j}\\ =20.0\widehat{i}-30.0\widehat{j}\end{array}$

The given vector can be represented in the form of $\overrightarrow{A}=A_x\widehat{i}+A_y\widehat{j}$as,

$\overrightarrow{A}=20.0\widehat{i}+\left(-30.0\right)\widehat{j}$

The components of the given vector A_x and A_y can be represented as,

A_x = 20.0

A_y = -30.0

Thus, the component of the given vector in the x-direction is A_x = 20.0 and the component in the y-direction is A_y = -30.0