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Question: Let \(\overset{\rightarrow}{u}\), \(\overset{\rightarrow}{v}\)and \(\overset{\rightarrow}{w}\) are v...

Let u→\overset{\rightarrow}{u}, v→\overset{\rightarrow}{v}and w→\overset{\rightarrow}{w} are vectors such that u→\overset{\rightarrow}{u}+ v→\overset{\rightarrow}{v}+w→\overset{\rightarrow}{w}=0→\overset{\rightarrow}{0}. If |u→\overset{\rightarrow}{u}|= 3, |v→\overset{\rightarrow}{v}|= 4 and |w→\overset{\rightarrow}{w}|= 5, then the value of

u→\overset{\rightarrow}{u}.v→\overset{\rightarrow}{v}+v→\overset{\rightarrow}{v} .w→\overset{\rightarrow}{w}+ w→\overset{\rightarrow}{w}.u→\overset{\rightarrow}{u}is-

A

–25

B

–27

C

28

D

25

Answer

–25

Explanation

Solution

Given,

u→\overset{\rightarrow}{u}+ v→\overset{\rightarrow}{v} + w→\overset{\rightarrow}{w}= 0→\overset{\rightarrow}{0}

If |u→\overset{\rightarrow}{u}|= 3; | v→\overset{\rightarrow}{v}| = 4 and |w→\overset{\rightarrow}{w}| = 5

Means u→\overset{\rightarrow}{u}, v→\overset{\rightarrow}{v} and w→\overset{\rightarrow}{w}form a right angled triangle, right angled at the angle between u→\overset{\rightarrow}{u}and v→\overset{\rightarrow}{v}

\ u→\overset{\rightarrow}{u}. v→\overset{\rightarrow}{v}= 0;

Further u→\overset{\rightarrow}{u}. w→\overset{\rightarrow}{w}=u→\overset{\rightarrow}{u} |w→\overset{\rightarrow}{w}|cos q,

but, cos q = –35\frac{3}{5}

u→\overset{\rightarrow}{u}. w→\overset{\rightarrow}{w} = (3) (5) (−35)\left( - \frac{3}{5} \right)= – 9

v→\overset{\rightarrow}{v}. w→\overset{\rightarrow}{w}=v→\overset{\rightarrow}{v} |w→\overset{\rightarrow}{w}|cos f

but cos f = –45\frac{4}{5}

v→\overset{\rightarrow}{v}. w→\overset{\rightarrow}{w} = (4) (5) (−45)\left( - \frac{4}{5} \right)= – 16

̃u→\overset{\rightarrow}{u} . v→\overset{\rightarrow}{v}+v→\overset{\rightarrow}{v}.w→\overset{\rightarrow}{w}+u→\overset{\rightarrow}{u}. w→\overset{\rightarrow}{w}= 0 – 16 – 9 = – 25