Question

Three vectors $\overrightarrow{A},\overrightarrow{B}\text{and}\overrightarrow{C}$ satisfy the relation $\overrightarrow{A}.\overrightarrow{B}=0$ and $\overrightarrow{A}.\overrightarrow{C}=0$. To which of the following vectors, The vector $\overrightarrow{A}$ is parallel to

• A. $\overrightarrow{B}$
• B. $\overrightarrow{C}$
• C. $\overrightarrow{B}.\overrightarrow{C}$
• D. $\overrightarrow{B}\times \overrightarrow{C}$

Answer 1

The correct option is D

$\overrightarrow{B}\times \overrightarrow{C}$

Step 1. Given Data,

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

$\overrightarrow{A}.\overrightarrow{C}=0$
Step 2. Formula Used,
We can write the scalar product of two vectors$\overrightarrow{a}$, and $\overrightarrow{b}$ as
$\overrightarrow{a}.\overrightarrow{b}=\left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta$
Where $\left|\overrightarrow{a}\right|$ and $\left|\overrightarrow{b}\right|$ is the magnitude of $\overrightarrow{a}$ and $\overrightarrow{b}$ respectively and $\theta$ is the angle between them.
When any of the vectors is zero or both vectors are perpendicular to each other, then the dot product of any two vectors is zero.

Step 3. Calculation,
According to the question, $\overrightarrow{A}.\overrightarrow{B}=0$ and $\overrightarrow{A}.\overrightarrow{C}=0$
Therefore, we can conclude that $\overrightarrow{A}$ is perpendicular to $\overrightarrow{B}$ and $\overrightarrow{A}$ is also perpendicular to $\overrightarrow{C}$. Mathematically,
$\overrightarrow{A}\mathrm{\perp }\overrightarrow{B}$
$\overrightarrow{A}\mathrm{\perp }\overrightarrow{C}$
The cross-product of two vectors gives resultant vectors perpendicular to both vectors.
Therefore, the resultant vector of the cross product of vector $\overrightarrow{B}$ and vector $\overrightarrow{C}$ will be perpendicular to both vectors, $\overrightarrow{B}$ and $\overrightarrow{C}$. Since vector $\overrightarrow{A}$ is also perpendicular to both vectors $\overrightarrow{B}$ and$\overrightarrow{C}$, it will lie in the same plane as the cross product$\overrightarrow{B}\times \overrightarrow{C}$.
From the above statement, we can conclude that $\overrightarrow{A}$ is parallel/antiparallel to $\overrightarrow{B}\times \overrightarrow{C}$.
Hence, Option D is the correct answer.