Question

Let \[a,b\] and \[c\] be the distinct non- negative numbers. If the vectors \[ai+aj+ck,i+k\] and \[ci+cj+bk\] lie in a plane, then \[c\] is?
A. The harmonic mean of \[a\] and \[b\]
B. Equal to zero
C. The arithmetic mean of \[a\] and \[b\]
D. The geometric mean of \[a\] and \[b\]

Answer 1

Hint: To solve this problem, first apply the condition when all the three of them lie in a plane. After applying that condition, try to simplify the matrix by performing some operations on it and then observe the condition of the geometric mean and you will get your required answer.

Complete step by step answer:
Before solving the given problem, let’s understand the concept of vectors.
Vector can be defined as a quantity which has both magnitude and direction. The vector value, in contrast to the scalar, has a directional and magnitude that helps determine the position of one point relative to another. It is also called the Euclidean vector, geometric vector and spatial vector.
The length section of the directed line is called the vector’s magnitude and the angle at which the vector is tilted indicates the direction of the vector. A vector consists of two elements, mainly horizontal and vertical. The value of the horizontal part is \[\cos \theta \] while the value of vertical part is \[\sin \theta \]
There are different types of vectors such as: Zero vector, Unit vector, Position vector, Co initial vector, Like and unlike vectors, Coplanar vectors, Collinear vectors, Equal vector, Displacement vector and Negative of a vector.
Now, let’s understand matrix:
A matrix can be defined as a rectangular array of numbers that are generally arranged in rows and columns (it can also be explained as an arrangement of certain quantities). If a matrix is defined as \[m\times n\] means that matrix has \[m\] rows (i.e. horizontal lines) and \[n\] columns (i.e. vertical lines).
The elements of a matrix may be real or complex numbers. If all the elements of a matrix are real, then the matrix is called a real matrix.
Types of matrices are as follows: Row Matrix, Column Matrix, Null Matrix, Square Matrix, Diagonal Matrix, Symmetric Matrix, Skew-Symmetric Matrix, Anti Symmetric Matrix etc.
Now, according to the given question:
Let, \[a=ai+aj+ck\] , \[b=i+k\] and \[c=ci+cj+bk\]
 \[a\] , \[b\] and \[c\] lies in a plane if:
\[\left[ a\text{ }b\text{ }c \right]=0\]
Substituting values in it as:
\[\Rightarrow \left| \begin{align}
  & a\text{ }a\text{ c} \\
 & 1\text{ 0 }1 \\
 & c\text{ }c\text{ }b \\
\end{align} \right|=0\]
Applying \[{{C}_{1}}\to {{C}_{1}}-{{C}_{2}}\] operation:
\[\Rightarrow \left| \begin{align}
  & \text{0 }a\text{ c} \\
 & 1\text{ 0 }1 \\
 & \text{0 }c\text{ }b \\
\end{align} \right|=0\]
\[\Rightarrow 1.[ab-{{c}^{2}}]=0\]
\[\Rightarrow ab-{{c}^{2}}=0\]
 \[\Rightarrow ab={{c}^{2}}\]
So, we can say that \[c\] is the geometric mean of \[a\] and \[b\]

So, the correct answer is “Option D”.

Note:
A matrix is said to be a symmetric matrix if it follows these given conditions: it has only real Eigenvalues, it is always diagonalizable and it has orthogonal Eigenvectors. And the sum and difference of any two symmetric matrices will always be symmetric but it is not always true in multiplication of matrices.