Question

If \[\overrightarrow{a}\], $\overrightarrow{b}$, and $\overrightarrow{c}$ are three coplanar vectors, then \[[\overrightarrow{a}+\overrightarrow{b}\text{ }\overrightarrow{b}+\overrightarrow{c}\text{ }\overrightarrow{c}+\overrightarrow{a}]=\]
A. \[[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]\]
B. \[2[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]\]
C. \[3[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]\]
D. \[0\]

Answer 1

Hint: In the above question, we are to find the value of the expression \[[\overrightarrow{a}+\overrightarrow{b}\text{ }\overrightarrow{b}+\overrightarrow{c}\text{ }\overrightarrow{c}+\overrightarrow{a}]\], which can easily be solved if we know the concept of coplanarity of vectors.

Formula used: In vector triple product is cross and dot products are interchangeable. i.e.,
\[\begin{align}
  & [\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]=\overrightarrow{a}\cdot \overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a}\times \overrightarrow{b}\cdot \overrightarrow{c}=\overrightarrow{b}\times \overrightarrow{c}\cdot \overrightarrow{a}=\overrightarrow{c}\times \overrightarrow{a}\cdot \overrightarrow{b} \\
 & [\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]=[\overrightarrow{b}\text{ }\overrightarrow{c}\text{ }\overrightarrow{a}]=[\overrightarrow{c}\text{ }\overrightarrow{a}\text{ }\overrightarrow{b}] \\
 & [\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]=-[\overrightarrow{b}\text{ }\overrightarrow{a}\text{ }\overrightarrow{c}]=-[\overrightarrow{c}\text{ }\overrightarrow{b}\text{ }\overrightarrow{a}]=-[\overrightarrow{a}\text{ }\overrightarrow{c}\text{ }\overrightarrow{b}] \\
\end{align}\]

Complete step by step solution: Here, we are given that the three vectors \[\overrightarrow{a}\], $\overrightarrow{b}$, and $\overrightarrow{c}$ are coplanar vectors, which means they all lie in the same plane.
The concept of coplanarity states that if the same vector lies in the same plane twice then the result of the corresponding determinant will be zero.
Here we will use the same concept and try to draw the outcome of the given expression.
\[[\overrightarrow{a}+\overrightarrow{b}\text{ }\overrightarrow{b}+\overrightarrow{c}\text{ }\overrightarrow{c}+\overrightarrow{a}]=[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]+[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{a}]+[\overrightarrow{a}\text{ }\overrightarrow{c}\text{ }\overrightarrow{c}]+[\overrightarrow{a}\text{ }\overrightarrow{c}\text{ }\overrightarrow{a}]+[\overrightarrow{b}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]+[\overrightarrow{b}\text{ }\overrightarrow{b}\text{ }\overrightarrow{a}]+[\overrightarrow{b}\text{ }\overrightarrow{c}\text{ }\overrightarrow{c}]+[\overrightarrow{b}\text{ }\overrightarrow{c}\text{ }\overrightarrow{a}]\]
We have drawn 8 different matrices from the given matrix by choosing the different possible pairs of elements.
Now, the matrices having the same elements twice will become zero as per the concept of coplanarity, and the resultant will be
\[\begin{align}
  & [\overrightarrow{a}+\overrightarrow{b}\text{ }\overrightarrow{b}+\overrightarrow{c}\text{ }\overrightarrow{c}+\overrightarrow{a}]=[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]+0+0+0+0+0+0+[\overrightarrow{b}\text{ }\overrightarrow{c}\text{ }\overrightarrow{a}] \\
 & \text{ }=[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]+[\overrightarrow{b}\text{ }\overrightarrow{c}\text{ }\overrightarrow{a}] \\
 & \text{ }=2[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}] \\
\end{align}\]
We know one more property of coplanar vector matrix that if the elements of the matrix are interchanged then the result will not change.
I.e.,
\[[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]=[\overrightarrow{b}\text{ }\overrightarrow{c}\text{ }\overrightarrow{a}]\]
So, the result of the given expression will be
\[[\overrightarrow{a}+\overrightarrow{b}\text{ }\overrightarrow{b}+\overrightarrow{c}\text{ }\overrightarrow{c}+\overrightarrow{a}]=2[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]\]
But the vectors \[\overrightarrow{a}\], $\overrightarrow{b}$, and $\overrightarrow{c}$ are coplanar vectors, \[[\overrightarrow{a}\text{ }\overrightarrow{b}\text{ }\overrightarrow{c}]=0\]
Therefore,
\[[\overrightarrow{a}+\overrightarrow{b}\text{ }\overrightarrow{b}+\overrightarrow{c}\text{ }\overrightarrow{c}+\overrightarrow{a}]=0\]

Thus, Option (D) is correct.

Additional Information:

Note: We can see that though the matrix has distinct elements the result will become zero because the coplanar vectors have determinants equal to 0.