Question

If the moduli of vectors $a,b,c$ are $3,4,5$ are 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. $\sqrt{12}$
• B. $12$
• C. $5\sqrt{2}$
• D. $50$

Answer 1

The correct option is C $5\sqrt{2}$

According to the given condition, we have

$a.(b+c)=0$ ....(1)

$b.(c+a)=0$ ....(2)

$c.(a+b)=0$ ....(3)

Now adding (1),(2) and (3), we get

$2(a.b+b.c+c.a)=0(\because a.b=b.aetc)$

Hence ${|a+b+c|}^2$

$=a^2+b^2+c^2+2(a.b+b.c+c.a)=3^2+4^2+5^2=9+16+25=50\Rightarrow |a+b+c|=\sqrt{50}=5\sqrt{2}$