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Let $x$ and $y$ be two vectors in a $3$ dimensional space and $<x,y>$ denote their 
dot product. Then the determinant
$det\begin{bmatrix}<x,x> & <x,y>\\ <y,x> & <y,y>\end{bmatrix}$

  1. is zero when $x$ and $y$ are linearly independent
  2. is positive when $x$ and $y$ are linearly independent
  3. is non-zero for all non-zero $x$ and $y$
  4. is zero only when either $x$ or $y$ is zero
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2 Answers

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< x,x > = x$^{2}$

< y,y > = y$^{2}$

< x,y > = <y,x > = xycos$\Theta$

--------------------------------------------------------------------

Now, determinant = x$^{2}$y$^{2}$ - (xycos$\Theta$)$^{2}$

                                  = x$^{2}$y$^{2}$ - x$^{2}$y$^{2}$cos$^{2}$$\Theta$

                                  = x$^{2}$y$^{2}$(1-cos$^{2}$$\Theta$)

                                  = x$^{2}$y$^{2}$(sin$^{2}$$\Theta$)

So from above equation determinant will be either zero or positive:

  1. Determinant will be zero when x and y are linearly dependent (i.e $\Theta$ = n$\pi$) or if either of x and y is zero.
  2. Determinant will be positive when x and y are linearly independent (i.e $\Theta$ $\neq$ n$\pi$)

So option B satisfies the condition.

 

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