Question

Let ${}^{\prime }{a}^{\prime }$ be a real number and $\overrightarrow{\alpha }=\hat{i}+2\hat{j},\overrightarrow{\beta }=2\hat{i}+a\hat{j}+10\hat{k},\overrightarrow{\gamma }=12\hat{i}+20\hat{j}+a\hat{k}$, be three vectors, then $\overrightarrow{\alpha },\overrightarrow{\beta }$ and $\overrightarrow{\gamma }$ are linearly independent for

• A. $a>0$
• B. $a<0$
• C. $a=0$
• D. no value of $a$

Answer 1

Answer: (A), (B), (C)

The correct options are
A $a>0$
B $a<0$
C $a=0$

$\left[\overrightarrow{\alpha }\overrightarrow{\beta }\overrightarrow{\gamma }\right]=\left|\begin{array}{ccc}1& 2& 0\\ 2& a& 10\\ 12& 20& a\end{array}\right|=a^2-24a+240>0$ for all $a$
$\therefore \overrightarrow{\alpha },\overrightarrow{\beta }\text{and}\overrightarrow{\gamma }$ are non-coplanar or linearly independent for all values of $a$