question_answer

A parallelogram is constructed on the vectors \[\overset{\to }{\mathop{\mathbf{a}}}\,=3\overset{\to }{\mathop{\alpha }}\,-\overset{\to }{\mathop{\beta }}\,,\,\,\overset{\to }{\mathop{\mathbf{b}}}\,=\overset{\to }{\mathop{\alpha }}\,+3\overset{\to }{\mathop{\beta }}\,\], if\[\left| \overset{\to }{\mathop{\alpha }}\, \right|=\left| \overset{\to }{\mathop{\beta }}\, \right|=2\]and angle between\[\overset{\to }{\mathop{\alpha }}\,\]and\[\overset{\to }{\mathop{\beta }}\,\]is\[\frac{\pi }{3}\], then length of a diagonal of the parallelogram is

A) \[4\sqrt{5}\]

B) \[4\sqrt{3}\]

C) \[4\sqrt{17}\]

D)  None of the above

Correct Answer: B

Solution :

Diagonal               \[\overrightarrow{\mathbf{AC}}=\overset{\to }{\mathop{\mathbf{a}}}\,+\overset{\to }{\mathop{\mathbf{b}}}\,\] \[\Rightarrow \]                               \[\left| \overrightarrow{\mathbf{AC}} \right|=\left| \overset{\to }{\mathop{\mathbf{a}}}\,+\overset{\to }{\mathop{\mathbf{b}}}\, \right|\] \[\Rightarrow \]               \[\left| \overrightarrow{\mathbf{AC}} \right|=\left| \overset{\to }{\mathop{\mathbf{a}}}\, \right|+\left| \overset{\to }{\mathop{\mathbf{b}}}\, \right|+2\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,\] \[\Rightarrow \]               \[{{\left| \overrightarrow{\mathbf{AC}} \right|}^{2}}=\{|3\overset{\to }{\mathop{\alpha }}\,-\overset{\to }{\mathop{\beta }}\,|+|\overset{\to }{\mathop{\alpha }}\,+3\overset{\to }{\mathop{\beta }}\,|\]                                                 \[+2(3\overset{\to }{\mathop{\alpha }}\,-\overset{\to }{\mathop{\beta }}\,)\cdot (\overset{\to }{\mathop{\alpha }}\,+3\overset{\to }{\mathop{\beta }}\,)\}\] \[=9\overset{\to }{\mathop{\alpha \,_{{}}^{2}}}\,+\overset{\to }{\mathop{\beta \,_{{}}^{2}}}\,-6\alpha \cdot \beta +\overset{\to }{\mathop{\alpha \,_{{}}^{2}}}\,+9\overset{\to }{\mathop{\beta \,_{{}}^{2}}}\,+6\overset{\to }{\mathop{\alpha }}\,\cdot \overset{\to }{\mathop{\beta }}\,\]                                 \[+6\overset{\to }{\mathop{\alpha \,_{{}}^{2}}}\,-6\overset{\to }{\mathop{\beta \,_{{}}^{2}}}\,+16\overset{\to }{\mathop{\alpha }}\,\cdot \overset{\to }{\mathop{\beta }}\,\] \[\Rightarrow \]               \[{{\left| \overrightarrow{\mathbf{AC}} \right|}^{2}}=16\overset{\to }{\mathop{\alpha \,_{{}}^{2}}}\,+4\overset{\to }{\mathop{\beta \,_{{}}^{2}}}\,+16\overset{\to }{\mathop{\alpha }}\,\cdot \overset{\to }{\mathop{\beta }}\,\] \[\Rightarrow \]               \[{{\left| \overrightarrow{\mathbf{AC}} \right|}^{2}}=64+16|\overset{\to }{\mathop{\alpha }}\,||\overset{\to }{\mathop{\beta }}\,|cos\frac{\pi }{3}\]                 \[{{\left| \overrightarrow{\mathbf{AC}} \right|}^{2}}=80+16\times 4\times \frac{1}{2}=112\] \[\Rightarrow \]               \[{{\left| \overrightarrow{\mathbf{AC}} \right|}^{2}}=\sqrt{112}=4\sqrt{7}\] Other diagonal is \[{{\left| \overrightarrow{\mathbf{BD}} \right|}^{2}}=|\overset{\to }{\mathop{\mathbf{a}}}\,-\overset{\to }{\mathop{\mathbf{b}}}\,|\] \[{{\left| \overrightarrow{\mathbf{BD}} \right|}^{2}}=|\overset{\to }{\mathop{\mathbf{a}}}\,{{|}^{2}}|\overset{\to }{\mathop{\mathbf{b}}}\,{{|}^{2}}-2|\overset{\to }{\mathop{\mathbf{a}}}\,|\cdot |\overset{\to }{\mathop{\mathbf{b}}}\,|\]                 \[=64+16-16\times 4\times \frac{1}{2}\]                 \[=80-32\]                 \[=48\] \[\Rightarrow \]               \[|\overrightarrow{\mathbf{BD}}|=\sqrt{48}=4\sqrt{3}\]