question_answer 59)

                  If \[|\vec{A}|=2\] and \[|\vec{B}|=4,\] then match the relations in column I with the angle \[\theta \] between A and B in column ? II.                

Column I	Column II
(a) \[\vec{A}.\,\vec{B}=0\]	(i) \[\theta =0\]
(b) \[\vec{A}.\vec{B}=+8\]	(ii) \[\theta =\,{{90}^{o}}\]
(c) \[\vec{A}.\vec{B}=4\]	(iii) \[\theta =\,{{180}^{o}}\]
(d) \[\vec{A}.\vec{B}=-8\]	(iv) \[\theta =\,{{60}^{o}}\]

                 

Answer:

                  (a) \[\vec{A}.\vec{B}\,=AB\,\cos \,\theta \,=0\] if \[\theta =\,{{90}^{o}}\]                 \[\therefore \] \[\,(a)\,\to \,(ii)\]                 (b) \[\vec{A}.\vec{B}\,=\,AB\,\,\cos \theta =\,8\] if \[\cos \,\,\theta =\,1\] or \[\theta \,=\,{{0}^{o}}\]                 \[\therefore \,(b)\,\,\to \,\,(i)\]                 (c) \[\vec{A}\,.\vec{B}\,=\,AB\,\,\cos \theta \,=\,4\] if \[\cos \theta \,=\,\frac{1}{2}\]or \[\theta \,=\,{{60}^{o}}\]                 \[\therefore \,\,(c)\,\,\to \,(iv)\] (d) \[\vec{A}\,.\vec{B}\,\,=\,AB\,\,\cos \,\theta \,=-8\] of \[\cos \theta =\,-1\] or \[\theta \,=\,{{180}^{0}}\]                 \[\therefore \,(d)\,\to \,(iii)\]