Question

If $\overrightarrow{x}\cdot \overrightarrow{a}=0$, $\overrightarrow{x}\cdot \overrightarrow{b}=0$ and $\overrightarrow{x}\cdot \overrightarrow{c}=0$ for some non-zero vector $\overrightarrow{x}$, then the true statement is
A. $[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]=0$
B. $[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]\ne 0$
C. $[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]=1$
D. None of these

Answer 1

Hint: In the above question, we need the basic idea about dot product, cross product and scalar triple product.

Formula used: The dot product of two vectors is
$\overrightarrow{a}\cdot \overrightarrow{b}=\left| \overrightarrow{a} \right|\left| \overrightarrow{b} \right|\cos (\overrightarrow{a},\overrightarrow{b})$
The cross-product of two vectors is
$\overrightarrow{a}\times \overrightarrow{b}=\left| \overrightarrow{a} \right|\left| \overrightarrow{b} \right|\sin (\overrightarrow{a},\overrightarrow{b})\overrightarrow{n}$

Complete step by step solution: Here, we are given some statements which give additional information about the vectors and their relationship with each other.
The information which we can derive from the given statements are:
Since the dot product of all three vectors is equal to zero, which means the angle between each one of them and the non-zero vector $\overrightarrow{x}$ is a right angle.
Also, at least one of the three given vectors \[\overrightarrow{a}\], \[\overrightarrow{b}\], and \[\overrightarrow{c}\] can have zero magnitudes or you can say that it is a zero vector.
In the first case, the three given vectors can also be coplanar or can be perpendicular to each other but here the second case will violate the given conditions as the relation of each of these vectors with the non-zero vector $\overrightarrow{x}$ is the same.
So, the three vectors \[\overrightarrow{a}\], \[\overrightarrow{b}\], and \[\overrightarrow{c}\] will be coplanar and it is represented by
$[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]=0$

Thus, Option (A) is correct.

Additional Information: In vector triple product, cross and dot products are interchangeable. i.e.,
\[\begin{align}
  & [\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]=\overrightarrow{a}\cdot \overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}\cdot \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{c}\cdot \overrightarrow{a}=\overrightarrow{c}\times \overrightarrow{a}\cdot \overrightarrow{b} \\
 & [\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]=[\overrightarrow{b}\text{ }\overrightarrow{c}\text{ }\overrightarrow{a}]=[\overrightarrow{c}\text{ }\overrightarrow{a}\text{ }\overrightarrow{b}] \\
 & [\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]=-[\overrightarrow{b}\text{ }\overrightarrow{a}\text{ }\overrightarrow{c}]=-[\overrightarrow{c}\text{ }\overrightarrow{b}\text{ }\overrightarrow{a}]=-[\overrightarrow{a}\text{ }\overrightarrow{c}\text{ }\overrightarrow{b}] \\
\end{align}\]
Important vector identities for solving vector equations are:
\[\overrightarrow{a}\times \overrightarrow{a}=0\]
\[[\overrightarrow{a}\text{ }\overrightarrow{a}\text{ }\overrightarrow{b}]=[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{a}]=[\overrightarrow{b}\text{ }\overrightarrow{a}\text{ }\overrightarrow{a}]=0\]
\[\begin{align}
  & \overrightarrow{i}\cdot \overrightarrow{i}=\overrightarrow{j}\cdot \overrightarrow{j}=\overrightarrow{k}\cdot \overrightarrow{k}=1 \\
 & \overrightarrow{i}\times \overrightarrow{j}=\overrightarrow{k} \\
 & \overrightarrow{j}\times \overrightarrow{k}=\overrightarrow{i} \\
 & \overrightarrow{k}\times \overrightarrow{i}=\overrightarrow{j} \\
\end{align}\]

Note: Here we can observe from the concept of the angle between two vectors and the vectors lying in the same plane or coplanarity, the relationship between the three given vectors.