# Given that $$\overrightarrow a . \overrightarrow b = 0$$ and $$\overrightarrow a$$ x $$\overrightarrow b = \overrightarrow 0$$ What can you conclude about the vectors $$\overrightarrow a$$ and $$\overrightarrow b$$?

$\begin{array}{1 1} (A) \overrightarrow a\:and\:\overrightarrow b \;are\; parallel \\ (B) \overrightarrow a\:and\:\overrightarrow b \;are\; \perp \\ (C) |\overrightarrow a|=0\:or\:|\overrightarrow b|= 0 \\ (D) |\overrightarrow a|=|\overrightarrow b|\end{array}$

Toolbox:
• $\overrightarrow a.\overrightarrow b=0$ if $\overrightarrow a\perp \overrightarrow b$
• $\overrightarrow a\times \overrightarrow b=0$ if $\overrightarrow a$ is parallel to $\overrightarrow b$
Step 1:
Given $\overrightarrow a.\overrightarrow b=0$ and $\overrightarrow a\times \overrightarrow b=0$
$\overrightarrow a.\overrightarrow b=0$ and also $\overrightarrow a\times \overrightarrow b=0$ implies that either $|\overrightarrow a|=0$ or $|\overrightarrow b|=0$
Step 2:
This is because $\overrightarrow a.\overrightarrow b=0$ if $\overrightarrow a\perp \overrightarrow b$ and $\overrightarrow a\times \overrightarrow b=0$ if $\overrightarrow a$ is parallel to $\overrightarrow b$
Two vectors cannot be both parallel and perpendicular.So either $|\overrightarrow a|=0$ or $|\overrightarrow b|=0$