# 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 $ ?

Answer

Verified

380.4k+ views

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 $.

Complete step-by-step answer:

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 $\overrightarrow a .\overrightarrow b = 0$if ‘$\left| a \right| = 0$’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) $\left| a \right| = 0$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 \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).

Complete step-by-step answer:

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 $\overrightarrow a .\overrightarrow b = 0$if ‘$\left| a \right| = 0$’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) $\left| a \right| = 0$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 \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).

Recently Updated Pages

Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts

The lightest gas is A nitrogen B helium C oxygen D class 11 chemistry CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Which place is known as the tea garden of India class 8 social science CBSE

What is pollution? How many types of pollution? Define it

Write a letter to the principal requesting him to grant class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Draw a diagram of a plant cell and label at least eight class 11 biology CBSE