Question

If vector a= x1i +y1j & vector b= x2i + y2j. Find the condition that would make vector a & vector b parallel to each other

Answer 1

Vector Analysis

Formula:

If $\mathrm{\vec{a}}$ and $\mathrm{\vec{b}}$ are two parallel vectors, then

$\quad\quad\mathrm{\vec{a}\times\vec{b}=\vec{0}}$.

$\quad\quad \mathrm{\hat{i}\times\hat{i}=\vec{0}}$

$\quad\quad \mathrm{\hat{j}\times\hat{j}=\vec{0}}$

$\quad\quad \mathrm{\hat{i}\times\hat{j}=-\hat{j}\times\hat{i}}$

Solution:

The given vectors are

$\quad\quad \mathrm{\vec{a}=x_{1}\hat{i}+y_{1}\hat{j}}$

$\quad\quad \mathrm{\vec{b}=x_{2}\hat{i}+y_{2}\hat{j}}$

Since $\mathrm{\vec{a}}$ and $\mathrm{\vec{b}}$ are parallel vectors,

$\quad \mathrm{\vec{a}\times\vec{b}=\vec{0}}$

$\Rightarrow \mathrm{(x_{1}\hat{i}+y_{1}\hat{j})\times (x_{2}\hat{i}+y_{2}\hat{j})=\vec{0}}$

$\Rightarrow \mathrm{x_{1}x_{2}(\hat{i}\times\hat{i})+x_{1}y_{2}(\hat{i}\times\hat{j})}\\ \quad \mathrm{+y_{1}x_{2}(\hat{j}\times\hat{i})+y_{1}y_{2}(\hat{j}\times\hat{j})=\vec{0}}$

$\Rightarrow \mathrm{x_{1}y_{2}(\hat{i}\times\hat{j})+y_{1}x_{2}(\hat{j}\times\hat{i})=\vec{0}}$

$\Rightarrow \mathrm{x_{1}y_{2}(\hat{i}\times\hat{j})-x_{2}y_{1}(\hat{i}\times\hat{j})=\vec{0}}$

$\Rightarrow \mathrm{(x_{1}y_{2}-x_{2}y_{1})(\hat{i}\times\hat{j})=\vec{0}}$

$\Rightarrow \boxed{\color{blue}{\mathrm{x_{1}y_{2}-x_{2}y_{1}=0}}}$

This is the required condition.