Q25DQ - University Physics with Modern Physics

Question

If $\overrightarrow{A}$.$\overrightarrow{B}$=0 does it necessarily follow that A = 0 or B= 0? Explain. (b) If $\overrightarrow{A}\times \overrightarrow{B}=0$ $\overrightarrow{A}$ = 0, does it necessarily follow that A = 0 or B = 0? Explain.

Step-by-Step Solution

Answer

Given-

$\overrightarrow{A}.\overrightarrow{B}=0$

$\overrightarrow{A}\times \overrightarrow{B}=0$

Step 1: To Prove

Necessarily follow that ,A=0 or B=0

Step 2: ( E xplanation , A → .   B → = 0 then it is necessary A=0,B=0) If A → = 0 , B → = 0

So, it doesn’t necessarily follow that A=0 or B=0 or it can be the case $\overrightarrow{A}$perpendicular $\overrightarrow{B}$.

Step 3: ( Explanation for A → × B → = 0 )

$\overrightarrow{A}\times \overrightarrow{B}=AB\sin \theta$

$\overrightarrow{A}\times \overrightarrow{B}=0, if \overrightarrow{A}=0 or \overrightarrow{B}=0$

Or  $\overrightarrow{A}, \overrightarrow{B}$ are in same direction or in opposite direction.

Hence it is not necessarily required for A and B =0 for  $\overrightarrow{A}\times \overrightarrow{B}=0$

(Conclusion)

For

$\left(\overrightarrow{A}.\overrightarrow{B }=0\right)$

$\left(\overrightarrow{A}\times \overrightarrow{B}=0\right)$

It is doesn’t necessarily required A=0 or B=0.