Vector Algebra Question 80
Question: $ \overset{\to }{\mathop{a}},,\overset{\to }{\mathop{b}},and\vec{c} $ are three vectors with magnitude $ |\overset{\to }{\mathop{a}},|=4,|\overset{\to }{\mathop{b}},|=4,|\overset{\to }{\mathop{c}},|=2 $ and such that $ \overset{\to }{\mathop{a}}, $ is perpendicular to is perpendicular to $ (\overset{\to }{\mathop{b}},+\overset{\to }{\mathop{c}},),\overset{\to }{\mathop{b}}, $ is perpendicular to $ (\overset{\to }{\mathop{c}},+\overset{\to }{\mathop{a}},) $ and $ \overset{\to }{\mathop{c}}, $ is perpendicular to $ (\overset{\to }{\mathop{a}},+\overset{\to }{\mathop{b}},) $ . It follows that $ |\overset{\to }{\mathop{a}},+\overset{\to }{\mathop{b}},+\overset{\to }{\mathop{c}},| $ is equal to:
Options:
A) 9
B) 6
C) 5
D) 4
Show Answer
Answer:
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 $
BETA
