A) \[{{0}^{o}}\]
B) \[{{30}^{o}}\]
C) \[{{90}^{o}}\]
D) \[{{30}^{o}}\]
Correct Answer: D
Solution :
Given that,\[|\overset{\to }{\mathop{\mathbf{a}}}\,|=3,\,\,|\overset{\to }{\mathop{\mathbf{b}}}\,|=5,\,\,|\overset{\to }{\mathop{\mathbf{c}}}\,|=7\]and \[\overset{\to }{\mathop{\mathbf{a}}}\,+\overset{\to }{\mathop{\mathbf{b}}}\,+\overset{\to }{\mathop{\mathbf{c}}}\,=0\Rightarrow \overset{\to }{\mathop{\mathbf{a}}}\,+\overset{\to }{\mathop{\mathbf{b}}}\,=-\overset{\to }{\mathop{\mathbf{c}}}\,\] \[\Rightarrow \] \[{{(\overset{\to }{\mathop{\mathbf{a}}}\,+\overset{\to }{\mathop{\mathbf{b}}}\,)}^{2}}={{(-\overset{\to }{\mathop{\mathbf{c}}}\,)}^{2}}\] \[\Rightarrow \] \[\overset{\to }{\mathop{{{\mathbf{a}}^{\mathbf{2}}}}}\,+\overset{\to }{\mathop{{{\mathbf{b}}^{\mathbf{2}}}}}\,+2\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,=\overset{\to }{\mathop{\mathbf{c}}}\,\] \[\Rightarrow \] \[|\overset{\to }{\mathop{\mathbf{a}}}\,{{|}^{2}}+|\overset{\to }{\mathop{\mathbf{b}}}\,{{|}^{2}}+2|\overset{\to }{\mathop{\mathbf{a}}}\,||\overset{\to }{\mathop{\mathbf{b}}}\,|\cos \theta =|\overset{\to }{\mathop{\mathbf{c}}}\,{{|}^{2}}\] \[\Rightarrow \] \[{{3}^{2}}+{{5}^{2}}+2(3)(5)\cos \theta ={{7}^{2}}\] \[\Rightarrow \] \[30\cos \theta =49-34=15\] \[\Rightarrow \] \[\cos \theta =\frac{1}{2}\Rightarrow \theta ={{60}^{o}}\] Note: If two vectors \[\overset{\to }{\mathop{\mathbf{a}}}\,\] and \[\overset{\to }{\mathop{\mathbf{b}}}\,\] are perpendicular, then \[\overset{\to }{\mathop{\mathbf{a}}}\,\cdot \overset{\to }{\mathop{\mathbf{b}}}\,=0\] and if two vectors are parallel, then \[\overset{\to }{\mathop{\mathbf{a}}}\,\times \overset{\to }{\mathop{\mathbf{b}}}\,=0\]
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