A) 9
B) 6
C) 5
D) 4
Correct Answer: B
Solution :
[b] Since, \[\overset{\to }{\mathop{a}}\,,\overset{\to }{\mathop{b}}\,\] and \[\overset{\to }{\mathop{c}}\,\] are three vectors with magnitude \[|\overset{\to }{\mathop{a}}\,|=|\overset{\to }{\mathop{b}}\,|=4\] and \[|\overset{\to }{\mathop{c}}\,|=2,\] |
As \[\overset{\to }{\mathop{a}}\,\] is perpendicular to \[(\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,)\] |
\[\Rightarrow \overset{\to }{\mathop{a}}\,.(\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,)=0\] or \[\overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{c}}\,)=0\] ?(i) |
\[\overset{\to }{\mathop{b}}\,\] is perpendicular to \[(\overset{\to }{\mathop{c}}\,+\overset{\to }{\mathop{a}}\,)\] |
\[\Rightarrow \overset{\to }{\mathop{b}}\,.(\overset{\to }{\mathop{c}}\,+\overset{\to }{\mathop{a}}\,)=0\] or \[\overset{\to }{\mathop{b}}\,.\overset{\to }{\mathop{c}}\,+\overset{\to }{\mathop{b}}\,.\overset{\to }{\mathop{a}}\,=0\] ?(ii) |
\[\Rightarrow \overset{\to }{\mathop{c}}\,\] is perpendicular to \[(\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,)\] |
\[\overset{\to }{\mathop{c}}\,.(\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,)=0\] or \[\overset{\to }{\mathop{c}}\,.\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{c}}\,.\overset{\to }{\mathop{b}}\,=0\] ?(iii) |
From equations (i), (ii) and (iii), we get |
\[\Rightarrow 2(\overset{\to }{\mathop{a}}\,.\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{b}}\,.\overset{\to }{\mathop{c}}\,+\overset{\to }{\mathop{c}}\,.\overset{\to }{\mathop{a}}\,)=0\] |
Further we know that |
\[|\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,{{|}^{2}}=|\overset{\to }{\mathop{a}}\,{{|}^{2}}+|\overset{\to }{\mathop{b}}\,{{|}^{2}}+|\overset{\to }{\mathop{c}}\,{{|}^{2}}\] |
\[+\overrightarrow{2a}.\vec{b}+\overrightarrow{2b}.\overrightarrow{c}+\overrightarrow{2c}.\overrightarrow{a}\] |
\[\Rightarrow |\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,{{|}^{2}}={{4}^{2}}+{{4}^{2}}+{{2}^{2}}+0=36\] |
or \[|\overset{\to }{\mathop{a}}\,+\overset{\to }{\mathop{b}}\,+\overset{\to }{\mathop{c}}\,|=6\] |
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