Answer
(a)$\theta = 97.123^{\circ}$
(b)$\theta = 90^{\circ}$
Work Step by Step
(a)$\vec A =4\vec i - 2 \vec j $
$\vec B =-2 \vec i - 3 \vec j $
$ \vec A. \vec B = |A||B|cos\theta $
$cos\theta = \frac { \vec A. \vec B }{ |A||B|}$
$|A| =\sqrt {{4^2}+{ 2^2}} = \sqrt{20} $
$|B| = \sqrt{{2^2} + {3^2}} = \sqrt{13} $
$ \vec A. \vec B = (4\vec i - 2 \vec j ).(-2 \vec i - 3 \vec j )$
$ \vec A. \vec B = - 8 + 6 = - 2$
$cos\theta = \frac { \vec A. \vec B }{ |A||B|}$
$cos\theta = \frac { -2}{\sqrt{13}× \sqrt{20}}$
$cos\theta= -0.124$
$\theta =cos^{-1}(-0.124)$
$\theta = 97.123^{\circ}$
(b) $\vec A =-4\vec i + 2 \vec j $
$\vec B =2 \vec i + 4 \vec j $
$ \vec A. \vec B = |A||B|cos\theta $
$cos\theta = \frac { \vec A. \vec B }{ |A||B|}$
$|A| =\sqrt {{4^2}+{ 2^2}} = \sqrt{20} $
$|B| = \sqrt{{4^2} + {2^2}} = \sqrt{20} $
$ \vec A. \vec B = (-4\vec i + 2 \vec j ).(2 \vec i +4 \vec j )$
$ \vec A. \vec B = - 8 + 8 = 0$
$cos\theta = \frac { \vec A. \vec B }{ |A||B|}$
$cos\theta = \frac { 0}{\sqrt{20}× \sqrt{20}}$
$cos\theta= 0$
$\theta =cos^{-1}(0)$
$\theta = 90^{\circ}$
