If a and b are non-zero vectors which are linearly dependent such… - Vidyadip

MCQ

If $\vec a$ and $\vec b$ are non-zero vectors which are linearly dependent such that $\frac{{\left| {\vec a + \vec b} \right|}}{{\left| {\vec a - \vec b} \right|}}\, = \,2,\,\left| {\vec b} \right|\, > \,\left| {\vec a} \right|$ Then

✓
$\vec b = 3\vec a$
B
$\vec b = - 3\vec a$
C
$\vec b = 2\vec a$
D
$\vec b = - 2\vec a$

✓

Answer

Correct option: A.

$\vec b = 3\vec a$

a
Let $\overrightarrow{\mathrm{b}}=\lambda \overrightarrow{\mathrm{a}}$

$ \Rightarrow \frac{{|\vec a + \lambda \vec a|}}{{|\vec a - \lambda \vec a|}} = 2\quad \Rightarrow \quad \frac{{1 + \lambda }}{{1 - \lambda }} = \pm 2$

$ \Rightarrow \lambda = 3,\frac{1}{3}\quad \Rightarrow \quad \vec b\left\langle {\begin{array}{*{20}{c}}
{3\vec a}\\
{\frac{{\vec a}}{3}}
\end{array}} \right.$

$ \Rightarrow \overrightarrow {\rm{b}} = 3\vec a\quad $ ($\because $ $|\overrightarrow {\rm{b}} | > |\overrightarrow {\rm{a}} |$)

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