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Question: If \|\(\overline{a}\)\| = 3, \|\(\overline{b}\)\| = 4, \|\(\overline{c}\)\| = 5 and \(\overline{a}\)...

If |a‾\overline{a}| = 3, |b‾\overline{b}| = 4, |c‾\overline{c}| = 5 and a‾\overline{a}^ (b‾\overline{b} +c‾\overline{c}),b‾\overline{b}^(c‾\overline{c} + a‾\overline{a}), c‾\overline{c}^

(a‾\overline{a}+b‾\overline{b}) then |a + b + c| is equal to-

A

50

B

25

C

525\sqrt{2}

D

12

Answer

525\sqrt{2}

Explanation

Solution

a‾\overline{a}^ (b‾\overline{b}+c‾\overline{c}) ̃a‾\overline{a} . (b‾\overline{b}+c‾\overline{c}) = 0

̃ a‾\overline{a} . b‾\overline{b} + a‾\overline{a}.c‾\overline{c} = 0 and two similar results

adding, 2(a‾\overline{a}.b‾\overline{b}+b‾\overline{b}.c‾\overline{c}+c‾\overline{c}.a‾\overline{a}) = 0

Now |a‾\overline{a} +b‾\overline{b} +c‾\overline{c} |2 = (a‾\overline{a}+ b‾\overline{b} +c‾\overline{c} ). (a‾\overline{a} + b‾\overline{b} +c‾\overline{c} )

= |a‾\overline{a}|2 + |b‾\overline{b}|2 + |c‾\overline{c}|2 + 2(a‾\overline{a}.b‾\overline{b} +b‾\overline{b} . c‾\overline{c} + c‾\overline{c}.a‾\overline{a})

= 9 + 16 + 25 + 0 = 50

\ | a‾\overline{a}+b‾\overline{b} +c‾\overline{c} | =525\sqrt{2}