ONE MORE STEP TOWARD BETTER TOMORROW : CHAPTER 7

Friday, 8 August 2014

CHAPTER 7 - Worked Out Examples

Example: 4

Suppose that for three non-zero vectors $\vec a,\vec b,\vec c,$ any two of them are non-collinear. If the vectors $\left( {\vec a + 2\vec b} \right)$ and $\vec c$ are collinear and the vectors $\left( {\vec b + 3\vec c} \right)$ and $\vec a$ are collinear, prove that

$\vec a + 2\vec b + 6\vec c = \vec 0$

Solution: 4

We must have some $\lambda ,\mu \in\mathbb{R}$ such that

	$\vec a + 2\vec b = \lambda \vec c$	$\ldots(1)$
	$\vec b + 3\vec c = \mu \vec a$	$\ldots(2)$

From $(1)$ , we have

$\vec c = \dfrac{1}{\lambda }\left( {\vec a + 2\vec b} \right)$

$\ldots(3)$

We use this in $(2)$ :

	$\vec b + \dfrac{3}{\lambda }\left( {\vec a + 2\vec b} \right) = \mu \vec a$
	$\Rightarrow \,\,\,\,\, \left( {\dfrac{3}{\lambda } - \mu } \right)\vec a + \left( {1 + \dfrac{6}{\lambda }} \right)\vec b = \vec 0$