Question

Angle between the given two vectors is 0°.

$\vec{u}=\alpha \hat{i} + 2\hat{j} + \hat{k}$

$\vec{v} = 2 \hat{i} + \alpha\hat{ j} + \beta \hat{k}$

Find the value of α and ß.​

Answer 1

Answer:

Check the attachment for Answer

Answer 2

$\large \dag$ Question :-

Angle between the given two vectors is 0°.

$\rm\vec{u}=\alpha \hat{i} + 2\hat{j} + \hat{k}$

$\rm\vec{v} = 2 \hat{i} + \alpha\hat{ j} + \beta \hat{k}$

Find the value of α and β.

$\large \dag$ Answer :-

$\red\dashrightarrow\underline{\underline{\sf \green{ \alpha = 2 \: and \: \beta = 1 \: or \: \alpha = - 2 \: and \: \beta = - 1 }} }$

$\large \dag$ Step by step Explanation :-

As angle b/w $\sf \large \vec u$ and $\sf \large \vec v$ is 0° so their cross product will be zero.

$:\longmapsto \rm \hat i(2 \beta - \alpha ) - \hat j( \alpha \beta - 2) + \hat k( { \alpha }^{2} - 4) = 0$

$:\longmapsto \rm 2 \beta - \alpha = 0 \: \: \: - - - - (1)$

$\rm Also \: \: \alpha \beta - 2 = 0 \: \: \: - - - - (2)$

$\rm Also \: \: { \alpha }^{2} - 4 = 0\: \: \: - - - - (3)$

From equation (3) we get :

$\\:\longmapsto \rm \alpha { }^{2} = 4$

$:\longmapsto \rm \alpha = \pm2$

Case 1 : When $\large \alpha = 2$

Putting value of α in equation (1) ;

$\\ :\longmapsto \rm 2 \beta -2 = 0$

$:\longmapsto \rm 2 \beta = 2$

$:\longmapsto \rm \beta = 1$

Case 2 : When $\large \alpha = -2$

Putting value of α in equation (1) ;

$\\ :\longmapsto \rm 2 \beta -(-2 )= 0$

$:\longmapsto \rm 2 \beta =- 2$

$:\longmapsto \rm \beta = -1$

¥ Therefore value of α and β are :

$\purple{ \underline {\boxed{{\bf \alpha = 2 \: and \: \beta = 1 \: or \: \alpha = - 2 \: and \: \beta = - 1} }}}$