Question

$a , b$ and $c$ are three vectors such that $| a |=1,| b |=2,| c |=3 \quad$ and $b , c $ are perpendicular. If projection of $b$ on $a$ is the same as the projection of $c$ on $a$, then $| a - b + c |$ is equal to

• A. $\sqrt{2}$
• B. $\sqrt{7}$
• C. $\sqrt{14}$
• D. $\sqrt{21}$

Answer 1

Answer: (C)

Given that,
$| a |=1,| b |=2,| c |=3$
$\because b$ and $c$ are perpendicular.
$\therefore b \cdot c=0\,\,\,...(i)$
And projection of $b$ on $a =\frac{ a \cdot b }{| a |}$
Projection of $c$ on $a =\frac{ a \cdot c }{| a |}$
$\because$ Both projections are same.
$\therefore \frac{ a \cdot b }{| a |}=\frac{ a \cdot c }{| a |}$
$\Rightarrow a \cdot b = a \cdot c$
Then $| a - b + c |^{2}=| a |^{2}+| b |^{2}+| c |^{2}-2 a \cdot b -2 b \cdot c +2 c \cdot a$
$=(1)^{2}+(2)^{2}+(3)^{2}-2 a \cdot b-0+2 a \cdot c$
$=1+4+9-2 a \cdot b+2 a \cdot b=14$
from Eqs. (i) and (ii)]
$\therefore | a - b + c |=\sqrt{14}$