91Ó°ÊÓ

The given vectors are$\stackrel{→}{A}=−3.00\widehat{i}+6.00\widehat{j}and\stackrel{→}{B}=7.00\widehat{i}+2.00\widehat{j}$.

Consider a vector quantity $\stackrel{→}{V}=V_x\widehat{i}+V_y\widehat{j}$ , Here V_x and V_y are the components along x, and y directions respectively and $\widehat{i},\widehat{j}$ are the unit vectors along x, and y directions respectively. The direction of this vector is expressed as,

$\tan \mathrm{θ}=\frac{V_y}{V_x}$

Part (a)

The expression for the direction of a vector is given by,

$\tan \mathrm{θ}=\frac{A_y}{A_x}$

Here, A_x and A_yare the components of $\stackrel{→}{A}$.

Substitute -3.00 for A_x , 6.00 for A_y.

$\begin{array}{c}\tan \mathrm{θ}=\frac{6.00}{−3.00}\\ =−2\\ \mathrm{θ}={\tan }^{−1}\left(−2\right)+180°\\ =116.6°\end{array}$

Thus, $\stackrel{→}{A}$ makes an angle of 116.6^o with x-axis.

Part (b)

The expression for the direction of a vector is given by,

$\tan \mathrm{θ}=\frac{B_y}{B_x}$

Here, B_y and B_x are the components of $\stackrel{→}{B}$.

Substitute 7.00 for B_x , 2.00 for B_y.

$\begin{array}{c}\tan \mathrm{θ}=\frac{2.00}{7.00}\\ =0.285\\ \mathrm{θ}={\tan }^{−1}\left(0.285\right)\\ =15.9°\end{array}$

Thus, $\stackrel{→}{B}$ makes an angle of 15.9^o with x-axis.

Part (c)

The sum of the given vectors is expressed as,

$\stackrel{→}{C}=\stackrel{→}{A}+\stackrel{→}{B}$

Substitute $−3.00\widehat{i}+6.00\widehat{j}$ for $\stackrel{→}{A}$, $7.00\widehat{i}+2.00\widehat{j}$ for $\stackrel{→}{B}$.

$\begin{array}{c}\stackrel{→}{C}=\left(−3.00\widehat{i}+6.00\widehat{j}\right)+\left(7.00\widehat{i}+2.00\widehat{j}\right)\\ =\left(−3.00+7.00\right)\widehat{i}+\left(6.00+2.00\right)\widehat{j}\\ =4.00\widehat{i}+8.00\widehat{j}\end{array}$

The expression for the direction of a vector is given by,

$\tan \mathrm{θ}=\frac{C_y}{C_x}$

Here, C_x and C_y are the components of $\stackrel{→}{C}$.

Substitute 4.00 for C_x , 8.00 for C_y .

$\begin{array}{c}\tan \mathrm{θ}=\frac{8.00}{4.00}\\ =2.00\\ \mathrm{θ}={\tan }^{−1}\left(2.00\right)\\ =63.4°\end{array}$

Thus, $\stackrel{→}{C}$ makes an angle of 63.4^o with x-axis.