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Question
If \[\vec{a} , \vec{b} , \vec{c}\] are any three mutually perpendicular vectors of equal magnitude a, then \[\left| \vec{a} + \vec{b} + \vec{c} \right|\] is equal to
Options
(a) a
(b) \[\sqrt{2}a\]
(c) \[\sqrt{3}a\]
(d) 2a
(e) None of these
Solution
(c) \[\sqrt{3}a\]
\[\text{ Given that }\]
\[\text{ So },\left| \vec{a} \right|=\left| \vec{b} \right|=\left| \vec{c} \right|=a . . . \left( i \right)\]
\[\text{ Since they are mutually perpendicular },\]
\[ \vec{a} . \vec{b} = \vec{b} . \vec{c} = \vec{c} . \vec{a} = 0 . . . \left( ii \right)\]
\[\text{ Now },\]
\[ \left| \vec{a} + \vec{b} + \vec{c} \right|^2 = \left| \vec{a} \right|^2 + \left| \vec{b} \right|^2 + \left| \vec{c} \right|^2 + 2 \vec{a} . \vec{b} + 2 \vec{b} . \vec{c} + 2 \vec{c} . \vec{a} \]
\[ = a^2 + a^2 + a^2 + 0 + 0 + 0 \left[ \text{ Using } \left( i \right) \text{ and } \left( ii \right) \right]\]
\[ = 3 a^2 \]
\[ \therefore \left| \vec{a} + \vec{b} + \vec{c} \right| = \sqrt{3}a\]
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