< x,x > = x$^{2}$
< y,y > = y$^{2}$
< x,y > = <y,x > = xycos$\Theta$
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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:
- 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.
- Determinant will be positive when x and y are linearly independent (i.e $\Theta$ $\neq$ n$\pi$)
So option B satisfies the condition.