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

Last updated date: 28th Mar 2023
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Hint: Here, we need to draw a conclusion about the vectors $\overrightarrow a$ and $\overrightarrow b$from the statements $\overrightarrow a .\overrightarrow b = 0$ and $\overrightarrow a \times \overrightarrow b = 0$ by considering $\overrightarrow a .\overrightarrow b = \left| a \right|.\left| b \right|.\cos \theta$and $\overrightarrow a \times \overrightarrow b = \left| a \right|.\left| b \right|.\sin \theta$.

Given,
i. $\overrightarrow a .\overrightarrow b = 0$.
Here, $\overrightarrow a .\overrightarrow b = 0$ is the dot product of the vectors $\overrightarrow a$ and $\overrightarrow b$.As, we know the dot product of two vectors can be written as:
$\overrightarrow a .\overrightarrow b = \left| a \right|.\left| b \right|.\cos \theta \to (1)$
Where:
$\left| a \right|$ Is the magnitude of$\overrightarrow a$, $\left| b \right|$is the magnitude of $\overrightarrow b$and $\theta$ is the angle between $\overrightarrow a$ and $\overrightarrow b$.
It is given that $\overrightarrow a .\overrightarrow b = 0$ i.e..,

$\left| a \right|.\left| b \right|.\cos \theta = 0 \to (2)$
So, from equation (2) we can say that the dot product of vectors $\overrightarrow a$ and $\overrightarrow b$is ‘0’ in the following cases.
(i) $\left| a \right| = 0$i.e.., the magnitude of $\overrightarrow a$is zero.
(ii) $\left| b \right| = 0$i.e.., the magnitude of $\overrightarrow b$is zero.
(iii) $\overrightarrow a \bot \overrightarrow b$i.e.., the angle between the vectors is${90^o}$$[\because \cos {90^o} = 0]. Hence, we can conclude that \overrightarrow a .\overrightarrow b = 0if ‘\left| a \right| = 0’or if ‘\left| b \right| = 0’or ‘if the vectors are perpendicular to each other. ii. \overrightarrow a \times \overrightarrow b = 0. Here, \overrightarrow a \times \overrightarrow b = 0 is the cross product of the vectors \overrightarrow a and \overrightarrow b .As, we know the cross product of two vectors can be written as: \overrightarrow a \times \overrightarrow b = \left| a \right|.\left| b \right|.\sin \theta \to (1) Where: \left| a \right| Is the magnitude of\overrightarrow a , \left| b \right|is the magnitude of \overrightarrow b and \theta is the angle between \overrightarrow a and \overrightarrow b . It is given that \overrightarrow a \times \overrightarrow b = 0 i.e.., \left| a \right|.\left| b \right|.\sin \theta = 0 \to (2) So, from equation (2) we can say that the cross product of vectors \overrightarrow a and \overrightarrow b is ‘0’ in the following cases (i) \left| a \right| = 0i.e.., the magnitude of \overrightarrow a is zero. (ii) \left| b \right| = 0i.e.., the magnitude of \overrightarrow b is zero. (iii)\overrightarrow a \parallel \overrightarrow b i.e.., the angle between the vectors is{0^o}$$[\because \sin {0^o} = 0]$.
Hence, we can conclude that $\overrightarrow a \times \overrightarrow b = 0$if ‘$\left| a \right| = 0$’or if ‘$\left| b \right| = 0$’or ‘if the vectors are parallel to each other.

Note: The dot product of two vectors will be $'0'$ if the vectors are perpendicular to each other (in case vectors are non-zero).Similarly, the cross product of two vectors will be $'0'$ if the vectors are parallel to each other (in case vectors are non-zero).