Question

A point charge $q$ is placed at origin. Let ${\vec E_A}$, ${\vec E_B}$ and ${\vec E_C}$ be the electric field at three points $A\left( {1,2,3} \right)$ , $B\left( {1,1, - 1} \right)$ and $C\left( {2,2,2} \right)$ due to charge $q$. Then,
(A) ${\vec E_A} \bot {\vec E_B}$
(B) ${\vec E_A}\parallel {\vec E_C}$
(C) $\left| {{{\vec E}_B}} \right| = 4\left| {{{\vec E}_C}} \right|$
(D) ${\vec E_B} = 8\left| {{{\vec E}_C}} \right|$

Answer 1

Hint: First the given point is written in the form of the vector equation, then by using the vector equation, the conditions which are given in the option can be checked. If both sides are equal, then the conditions are satisfied. There are some conditions for perpendicular and parallel of the vector equations.

Complete step by step solution
Given that,
The point of $A$ is, $A\left( {1,2,3} \right)$,
The point of $B$ is $B\left( {1,1, - 1} \right)$,
The point of $C$ is $C\left( {2,2,2} \right)$.
The vector form of the point are as follows,
The vector equation of the point $A$ is, ${\vec E_A} = \hat i + 2\hat j + 3\hat k$
The vector equation of the point $B$ is, ${\vec E_B} = \hat i + \hat j - \hat k$
The vector equation of the point $C$ is, ${\vec E_C} = 2\hat i + 2\hat j + 2\hat k$
Now check the conditions which are given in the options.
1. ${\vec E_A} \bot {\vec E_B}$
It is given that the vector equation of $A$ and vector equation of $B$, to check the two vectors are perpendicular, then their dot product must be equal to zero, then
${\vec E_A}.{\vec E_B} = 0$
By substituting the vector equation of $A$ and the vector equation of $B$ in the above equation, then the above equation is written as,
$\left( {\hat i + 2\hat j + 3\hat k} \right).\left( {\hat i + \hat j - \hat k} \right) = 0$
In the vector multiplication, the coefficient of the $\hat i$ of one equation is multiplied with the coefficient of the $\hat i$ of the other equation, it does not multiplied with the coefficient of the other terms. Then, by multiplying the terms in the above equation, then
$1 + 2 - 3 = 0$
In the above equation, the coefficient of $\hat i$ is multiplied with the coefficient of $\hat i$ and the coefficient of $\hat j$ is multiplied with the coefficient of $\hat j$ and the coefficient of $\hat k$ is multiplied with the coefficient of $\hat k$. Then, by adding the terms in the above equation, then
$3 - 3 = 0$
By subtracting the terms in the above equation, then
$0 = 0$
Here, the condition is satisfied, So, the vector equation of $A$ and the vector equation of $B$ are perpendicular to each other.

Hence, the option (A) is the correct answer.

Note: If the two vectors are in parallel, then the cross product of the two vectors must be equal to zero. The cross product of the vector equation is done by making the matrix equation of the two vectors and the cross product is done. If the result is zero then the two vectors are said to be parallel.