A) \[\frac{19}{2}\]
B) 8
C) \[\frac{17}{2}\]
D) 9
Correct Answer: A
Solution :
Given \[\overrightarrow{a}=\widehat{i}-\widehat{j}\] ?.. (1) \[\overrightarrow{b}\,\,=\,\,\widehat{i}\,+\,\,\widehat{j}\,\,+\,\,\widehat{k}\] ?.. (2) \[\overrightarrow{a}\,\,.\,\,\overrightarrow{c}\,\,=\,\,4\] ...... (3) \[\left| \,\overrightarrow{\text{a}}\, \right|\text{ }\,\,\left| \,\overrightarrow{c}\, \right|\text{ }cos\,\theta \,\,=\,\,4\] \[\cos \,\,\theta \,\,=\,\,\frac{2\sqrt{2}}{\left| \overrightarrow{c} \right|}\] Also \[\overrightarrow{a}\,\,\times \,\,\overrightarrow{c}\,\,+\,\,\overrightarrow{b}\,\,=\,\,0\] \[\overrightarrow{a}\,\,\times \,\,\overrightarrow{c}\,\,=\,\,-\overrightarrow{b}\] \[\,\left| \overrightarrow{a}\,\,\,\,\times \,\,\,\,\,\overrightarrow{c} \right|\,\,\,=\,\,\,\left| \overrightarrow{b} \right|\] \[\left| \overrightarrow{a} \right|\,\,\,\,\,\,\left| \overrightarrow{c} \right|\,\,\,\sin \,\,\theta \,\,=\,\,\,\left| \overrightarrow{b} \right|\] \[\sqrt{2}\,\times \,\,\left| \,\overrightarrow{c}\, \right|\,\,\,sin\,\theta \,\,=\,\,\sqrt{3}\] \[\sqrt{2}\,\,\left| \,\overrightarrow{c}\, \right|\,\,\,\sqrt{1\,\,-\,\,{{\cos }^{2}}\,\theta }\,\,=\,\,\sqrt{3}\] Squaring \[2\,\,\left| \overrightarrow{c} \right|\,{{\,}^{2}}\,\left( 1-\frac{8}{{{\left| \overrightarrow{c} \right|}^{2}}} \right)\,\,\,=\,\,\,3\] \[2\,\,\left| \,\overrightarrow{c}\, \right|\,{{\,}^{2}}\,-16\,\,\,=\,\,\,3\] \[\,\left| \,\overrightarrow{c}\, \right|\,{{\,}^{2}}\,=\,\,\frac{19}{2}\]
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