Vectors $\underset{A}{\rightarrow},\underset{B}{\rightarrow}\:and\:\underset{C}{\rightarrow}$ are such that $\underset{A}{\rightarrow}.\underset{B}{\rightarrow}=0\:and\:\underset{A}{\rightarrow}.\underset{C}{\rightarrow}=0$. Then the vector parallel to $\underset{A}{\rightarrow}$ is Option 1) $\underset{B}{\rightarrow}\:and\:\underset{C}{\rightarrow}$ Option 2) $\underset{A}{\rightarrow}\times \underset{B}{\rightarrow}$ Option 3) $\underset{B}{\rightarrow}+ \underset{C}{\rightarrow}$ Option 4) $\underset{B}{\rightarrow}\times \underset{C}{\rightarrow}$

As learnt in

Vector or cross product -

Vector or cross product of two vector $\vec{A}$ & $\vec{B}$ written as$A\times B$ is a single vector whose magnitude is equal to product of magnitude of $\vec{A}$ & $\vec{B}$ and the sine of smaller angle$\Theta$  between them.

$\vec A\times \vec B= A\, B\sin \Theta$

- wherein

Figure 6 shows representation of vector or cross product of vectors.

shows representation of vector or cross product of vectors

and

Scalar , Dot or Inner Product -

Scalar product of two vector $\vec{A}$ & $\vec{B}$ written as $\vec{A}$ $\cdot$ $\vec{B}$ is a scalar quantity given by the product of magnitude of $\vec{A}$ & $\vec{B}$ and the cosine of smaller angle between them.

$\vec{A}\cdot \vec{B}= A\, B\cdot \cos \Theta$

- wherein

showing representation of scalar products of vectors.

$\vec{A}\times \left ( \vec{B} \times \vec{C}\right ) = \vec{B}\left ( \vec{A}\cdot \vec{C} \right ) - \vec{C}\left ( \vec{A} \cdot \vec{B}\right )=0$

$\Rightarrow \vec{A}\parallel \left ( \vec{B} \times \vec{C}\right )$

$\mid \because \vec{A}\cdot \vec{B} =0\ and\ \vec{A}\cdot \vec{C} =0$

$\left ( \vec{A}+\vec{B} \right )^{2}=\left ( \vec{C} \right )^{2}$

$A^{2}+B^{2}+2\vec{A}\vec{B} = C^{2}$

$3^{2}+4^{2}+2\vec{A}\cdot \vec{B} = 5^{2}$

$2\vec{A}\cdot \vec{B}=0\ \ \ \Rightarrow \vec{A}\cdot \vec{B} =0$

$\vec{A}\perp \vec{B}$

Option 1)

$\underset{B}{\rightarrow}\:and\:\underset{C}{\rightarrow}$

This option is incorrect

Option 2)

$\underset{A}{\rightarrow}\times \underset{B}{\rightarrow}$

This option is incorrect

Option 3)

$\underset{B}{\rightarrow}+ \underset{C}{\rightarrow}$

This option is incorrect

Option 4)

$\underset{B}{\rightarrow}\times \underset{C}{\rightarrow}$

This option is correct

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