Show that the vectors  4 bb ul a - 8bb ul b and -26bb ul a+52bb ul b  are parallel?

Dec 28, 2017

$\boldsymbol{\underline{u}}$ and $\boldsymbol{\underline{v}}$ are parallel.

Explanation:

Let us assume that the vectors $\boldsymbol{\underline{u}}$ and $\boldsymbol{\underline{v}}$ are parallel then there exists $l a m \mathrm{da} \in \mathbb{R}$ such that:

$\boldsymbol{\underline{u}} = l a m \mathrm{da} \boldsymbol{\underline{v}}$

And using the definitions of $\boldsymbol{\underline{u}}$ and $\boldsymbol{\underline{v}}$ we have:

$4 \boldsymbol{\underline{a}} - 8 \boldsymbol{\underline{b}} = l a m \mathrm{da} \left(- 26 \boldsymbol{\underline{a}} + 52 \boldsymbol{\underline{b}}\right)$
$4 \boldsymbol{\underline{a}} - 8 \boldsymbol{\underline{b}} = - 26 l a m \mathrm{da} \boldsymbol{\underline{a}} + 52 l a m \mathrm{da} \boldsymbol{\underline{b}}$

Equating coefficients of $\boldsymbol{\underline{a}}$ and $\boldsymbol{\underline{b}}$

$4 \setminus \setminus \setminus = - 26 l a m \mathrm{da} \implies l a m \mathrm{da} = - \frac{2}{13}$
$- 8 = 52 l a m \mathrm{da} \setminus \setminus \setminus \setminus \implies l a m \mathrm{da} = - \frac{2}{13}$

Hence we have found a suitable $l a m \mathrm{da} = - \frac{2}{13} \in \mathbb{R}$ such that

$\boldsymbol{\underline{u}} = l a m \mathrm{da} \boldsymbol{\underline{v}}$

Hence $\boldsymbol{\underline{u}}$ and $\boldsymbol{\underline{v}}$ are parallel.