Question

If $a,b,c$ are three vectors such that $\mid a\mid =3,\mid b\mid =4,\mid c\mid =5$ and $a,b,c$ are perpendicular to $b+c,c+a,a+b$ respectively, then $\mid a+b+c\mid =?$

• A. $5\sqrt{2}$
• B. $6\sqrt{2}$
• C. $3\sqrt{2}$
• D. $4\sqrt{2}$

Answer 1

The correct option is A

$5\sqrt{2}$

Explanation for the correct option:

Ste 1. Find the value of $\mid a+b+c\mid$:

Given, $a,b,c$ are three vectors such that $\mid a\mid =3,\mid b\mid =4,\mid c\mid =5$ and

$a,b,c$ are perpendicular to $b+c,c+a,a+b$ respectively

that means,

$a\perp (b+c)$

$\Rightarrow$ $a·(b+c)=0$

$\Rightarrow$$a·b+a·c=0$ …(1)

Similarly,

$b\perp (c+a)$

$\Rightarrow$$b·c+b·a=0$ …(2)

and $c\perp (a+b)$

$\Rightarrow$$c·a+c·b=0$ …(3)

Step 2. Add equation (1), (2), and (3),

$2(a·b+b·c+c·a)=0$

$\begin{array}{rcl}\mathbf{\because }\mathbf{|}\mathbf{a}\mathbf{+}\mathbf{b}\mathbf{+}\mathbf{c}{\mathbf{|}}^{\mathbf{2}}\mathbf{}& \mathbf{=}& \mathbf{}\mathbf{|}\mathbf{a}{\mathbf{|}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{|}\mathbf{b}{\mathbf{|}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{|}\mathbf{c}{\mathbf{|}}^{\mathbf{2}}\mathbf{}\mathbf{+}\mathbf{}\mathbf{2}\mathbf{(}\mathbf{a}\mathbf{·}\mathbf{b}\mathbf{}\mathbf{+}\mathbf{}\mathbf{b}\mathbf{·}\mathbf{c}\mathbf{}\mathbf{+}\mathbf{}\mathbf{c}\mathbf{·}\mathbf{a}\mathbf{)}\\ & =& |a{|}^2+|b{|}^2+|c{|}^2+0\\ & =& 9+16+25\\ & =& 50\end{array}$

$\begin{array}{rcl}\therefore |a+b+c|& =& \sqrt{50}\\ & =& \mathbf{5}\sqrt{\mathbf{2}}\end{array}$

Hence, Option ‘A’ is Correct.