91Ó°ÊÓ

VISUALIZE

Each vector is expressed as the addition of two components, one component along the x-direction and the other component in the $y$-direction. So we can use the tip-to-tail rule for vector addition and draw the vectors $\stackrel{→}{A}$and $\stackrel{→}{B}$using the given components, as shown below.

SOLVE

Given that,

Vector,$\stackrel{→}{A}=4\widehat{i}−2\widehat{j}$

Vector,$\stackrel{→}{B}=−3\widehat{i}+5\widehat{j}$

(a) Given that the vector

$\stackrel{→}{D}=\stackrel{→}{A}−\stackrel{→}{B}$

Substitute for $\stackrel{→}{A}$and $\stackrel{→}{B}$from equations (1) and (2).

$\begin{array}{l}\stackrel{→}{D}=\stackrel{→}{A}−\stackrel{→}{B}\\ =(4\widehat{i}−2\widehat{j})−(−3\widehat{i}+5\widehat{j})\\ =7\widehat{i}−7\widehat{j}\end{array}$

Thus, the vector, $\stackrel{→}{D}=7\widehat{i}−7\widehat{j}$.

(b) We are given that $\stackrel{→}{D}=7\widehat{i}−7\widehat{j}$. Note that,

$\begin{array}{l}\stackrel{→}{D}=\stackrel{→}{A}−\stackrel{→}{B}\\ =\stackrel{→}{A}+(−\stackrel{→}{B})\\ =(4\widehat{i}−2\widehat{j})+(3\widehat{i}−5\widehat{j})\end{array}$

Now use the tip-to-tail rule for the vectors $\stackrel{→}{A}$and $−\stackrel{→}{B}$to obtain the vector $\stackrel{→}{D}$as shown in the below figure.

(c) From part (a), the resultant vector,$\stackrel{→}{D}=7\widehat{i}−7\widehat{j}$

Hence the magnitude of the vector${\stackrel{→}{D}}^{\text{is,}}$

$\begin{array}{l}\left|\stackrel{→}{D}\right|=\sqrt{7^2+{(−7)}^2}\\ =\sqrt{49+49}\\ =\sqrt{98}\end{array}$

And the direction of the vector $\stackrel{→}{D}$is given by,

$\begin{array}{l}\mathrm{θ}={\tan }^{−1}\left(\frac{−7}{7}\right)\\ ={\tan }^{−1}(−1)\\ =−45\end{array}$