If a, b, c are coplanar vectors,then | vedclass

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(B) Since $a, b,$ and $c$ are coplanar,there exist scalars $x, y, z$ (not all zero) such that $xa + yb + zc = 0$ $(i)$.Taking the dot product of $(i)$

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$\left| \begin{array}{ccc} a & b & c \ a \cdot a & a \cdot b & a \cdot c \ b \cdot a & b \cdot b & b \cdot c \end{array} ight| = 0$

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(B) Since $a, b,$ and $c$ are coplanar,there exist scalars $x, y, z$ (not all zero) such that $xa + yb + zc = 0$ $(i)$.Taking the dot product of $(i)$ with $a$,we get $x(a \cdot a) + y(a \cdot b) + z(a \cdot c) = 0$ $(ii)$.Taking the dot product of $(i)$ with $b$,we get $x(b \cdot a) + y(b \cdot b) + z(b \cdot c) = 0$ $(iii)$.Since $x, y, z$ are not all zero,the system of equations $(i), (ii),$ and $(iii)$ has a non-trivial solution.Therefore,the determinant of the coefficients must be zero:$\left| \begin{array}{ccc} a & b & c \ a \cdot a & a \cdot b & a \cdot c \ b \cdot a & b \cdot b & b \cdot c \end{array} ight| = 0$.

If $a, b, c$ are coplanar vectors,then

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