Question

If $\overline{x}\cdot \overline{y}=\overline{x}\cdot \overline{z}$ and $\overline{x}\times \overline{y}=\overline{x}\times \overline{z}$ and $\overline{x}\ne \overline{0}$, then prove that $\overline{y}=\overline{z}$.

Answer 1

Consider the problem

$\overline{x}\cdot \overline{y}=\overline{x}\cdot \overline{z}$

$\overline{x}\cdot \overline{y}-\overline{x}\cdot \overline{z}=0$

$x\cdot (\overrightarrow{y}-\overrightarrow{z})=0$ ------ $\left(1\right)$

$\overrightarrow{x}\times \overrightarrow{y}=\overrightarrow{x}\times \overrightarrow{z}$

$\overrightarrow{x}\times \overrightarrow{y}-\overrightarrow{x}\times \overrightarrow{z}=0$

$\overrightarrow{x}\times (\overrightarrow{y}-\overrightarrow{z})=0$ ----- $\left(2\right)$

from $\left(1\right)$ and $\left(2\right)$

$\overrightarrow{x}=0$ or $\overrightarrow{y}-\overrightarrow{z}=0$

$\overrightarrow{y}=\overrightarrow{z}$ $(\overrightarrow{x}\ne 0)$