question_answer

There are two displacement vectors, one of magnitude 3 metres and the other of 4 metres. How would the two vectors be added so that the magnitude of the resultant vector be (a) 7 metres(b) 1 metre and (c) 5 metres.

Answer:

                The magnitude of resultant vector \[\vec{R}\]of two vectors \[\vec{A}\] and \[\vec{B}\]is given by\[\text{R}=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }\]. Case (i). R will be maximum if \[\cos \theta =1\] or\[\theta =0{}^\circ \]. Then\[\text{R}=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB(1)}=\left( \text{A}+\text{B} \right)\]. Case (ii). R will be minimum if \[\text{cos}\theta =-\text{1}\]or\[\theta =\text{18}0{}^\circ \]. Then \[\text{R}=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB(-1)}=\left( \text{A-B} \right)\]or \[\left( \text{B}-\text{A} \right)\]