Question

If $a=i-k$ , $b=xi+j+\left( 1-x \right)k$ and $c=yi+xj+\left( 1+x-y \right)k$. Then, $\left[ a\text{ }b\text{ }c \right]$ depends on?
1. Neither $x$ nor $y$ .
2. Both $x$ and $y$ .
3. Only $x$ .
4. Only $y$ .

Answer 1

In this problem we need to check whether the value of $\left[ a\text{ }b\text{ }c \right]$ depends on the variables $x$ and $y$ . For this we will first calculate the value of $\left[ a\text{ }b\text{ }c \right]$ by using the values of given vectors $a$ , $b$ and $c$. We will simplify the obtained determinant by performing column operations or the row operations. From the simplified value of $\left[ a\text{ }b\text{ }c \right]$ we can check whether the value is depending on the variables $x$ and $y$.

Complete step by step answer:
Given values of the vectors are $a=i-k$ , $b=xi+j+\left( 1-x \right)k$ and $c=yi+xj+\left( 1+x-y \right)k$.
Now the value of $\left[ a\text{ }b\text{ }c \right]$ can be written as
$\left[ a\text{ }b\text{ }c \right]=\left| \begin{matrix} 1 & 0 & -1 \\\ x & 1 & 1-x \\\ y & x & 1+x-y \\\ \end{matrix} \right|$
Perform the column operation ${{C}_{3}}\to {{C}_{3}}+{{C}_{1}}$ in the above determinant, then we will have
$\begin{aligned} & \left[ a\text{ }b\text{ }c \right]=\left| \begin{matrix} 1 & 0 & 1-1 \\\ x & 1 & 1-x+x \\\ y & x & 1+x-y+y \\\ \end{matrix} \right| \\\ & \left[ a\text{ }b\text{ }c \right]=\left| \begin{matrix} 1 & 0 & 0 \\\ x & 1 & 1 \\\ y & x & 1+x \\\ \end{matrix} \right| \\\ \end{aligned}$
Now the above determinant can be simplified as
$\left[ a\text{ }b\text{ }c \right]=1\left| \begin{matrix} 1 & 1 \\\ x & 1+x \\\ \end{matrix} \right|$
Use the determinant formula $\left| \begin{matrix} a & b \\\ c & d \\\ \end{matrix} \right|=ad-bc$ in the above equation.
$\left[ a\text{ }b\text{ }c \right]=1\left[ 1\left( 1+x \right)-1\left( x \right) \right]$
Distribution law of multiplication says that $a\left( b+c \right)=ab+ac$ . Apply this formula in the above equation.
$\left[ a\text{ }b\text{ }c \right]=1\left[ 1+x-x \right]$
We know that the value of $x-x$ is equal to $0$ . Now the above equation is modified as
$\begin{aligned} & \left[ a\text{ }b\text{ }c \right]=1\left( 1 \right) \\\ & \Rightarrow \left[ a\text{ }b\text{ }c \right]=1 \\\ \end{aligned}$

The value of $\left[ a\text{ }b\text{ }c \right]$ is equal to $1$ and there is no presence of any variable like $x$ and $y$ in the value of $\left[ a\text{ }b\text{ }c \right]$ which shows that the value $\left[ a\text{ }b\text{ }c \right]$ doesn’t depend on the variables $x$ and $y$ even though the vectors $a$ , $b$ and $c$ are expressed in terms of $x$ and $y$.

So, the correct answer is “Option 1”.

Note: In this problem while solving the determinant we have used the column operation to simplify the determinant. We can also perform the row operations to simplify the determinant or we can also directly calculate the determinant value by using the formula.