question_answer 60)

                  If \[|A|=2\] and \[|B|=4,\] then match the relations in column I with the angle between A and B in column II                

Column I	Column II
(a) \[|\vec{A}\times \vec{B}|=0\]	(i) \[\theta ={{30}^{o}}\]
(b) \[|\vec{A}\times \vec{B}|=8\]	(ii) \[\theta ={{45}^{o}}\]
(c) \[|\vec{A}\times \vec{B}|=4\]	(iii) \[\theta ={{90}^{o}}\]
(d) \[|\vec{A}\times \vec{B}|=4\sqrt{2}\]	(iv) \[\theta ={{0}^{o}}\]

 

Answer:

                  (a) \[|\vec{A}\times \,\vec{B}|\,=AB\,\,\sin \,\theta =\,0\] if \[\sin \,\theta =0\] or \[\theta =\,{{0}^{o}}\]                 \[\therefore \,(a)\,\to \,(iv)\]                 (b) \[|\vec{A}\,\times \,\vec{B}|\,\,=AB\,\sin \,\theta =\,8\] if \[\sin \theta =1\] or \[\theta ={{90}^{o}}\]                 \[\therefore \,(b)\,\to \,\,(iii)\] (c) \[|\vec{A}\times \vec{B}|\,=\,AB\,\,\sin \theta =\,8\,\,\sin \theta =\,4\,\] if \[\sin \,\theta =\,\frac{1}{2}\]            or \[\theta =\,{{30}^{o}}\] \[\therefore \,(c)\,\,\to \,(i)\] (d) \[|\vec{A}\times \,\vec{B}|\,=8\,\,\sin \,\,\theta =4\sqrt{2}\] if \[\sin \,\theta \,=\,\frac{1}{\sqrt{2}}\] or \[\theta =\,{{45}^{o}}\]                 \[\therefore \,(d)\,\to \,(ii)\]