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Q.
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
Vector Algebra
Solution:
According to the given condition, we have
$a \cdot(b+c)=0\,\,\, (1)$
$b \cdot(c+a)=0\,\,\, (2)$
$c \cdot(a+b)=0\,\,\, (3)$
Now adding (1), (2) and (3), we get
$2(a \cdot b+b \cdot c+c \cdot a)=0 $
$(\because a \cdot b=b \cdot a$ etc. $)$
Hence, $|a+b+c|^{2}$
$=a^{2}+b^{2}+c^{2}+2$
$(a \cdot b+b \cdot c+c \cdot a)$
$=3^{2}+4^{2}+5^{2}$
$=9+16+25=50$
$\Rightarrow |a+b+c|$
$=\sqrt{50}=5 \sqrt{2}$