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Cross Product of Two Vectors

If the moduli...

Question

If the moduli of the vectors a, b, c are 3, 4, 5 respectively and a and b + c, b and c + a, c and a + b are mutually perpendicular, then the modulus of a + b + c is

A

√12

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B

12

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C

5√2

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D

50

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Solution

The correct option is C 5√2
According to the given comdition,
$a . (b + c) = 0 . . . . . (i)$
$b . (c + a) = 0 . . . . . (i i)$
$c . (a + b) = 0 . . . . . (i i i)$
Now adding (i),(ii) and (iii),we get
2(a.b+b.c+c.a)=0. ${∵$ a.b=b.a etc.}
Hence, $| a + b + c |^{2} = a^{2} + b^{2} + c^{2} + 2 (a . b + b . c + c . a)$
$= 3^{2} + 4^{2} + 5^{2}$
$\Rightarrow | a + b + c | = \sqrt{50} = 5 \sqrt{2} .$

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