Consider the vectors u = a hat{i} + b hat{j} | vedclass

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(D) The vectors $u, v, w$ are coplanar if and only if their scalar triple product is zero,which is equivalent to the determinant of the matrix formed

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either one of $a, b$ and $c$ is zero,or any two of $a, b$ and $c$ are equal

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(D) The vectors $u, v, w$ are coplanar if and only if their scalar triple product is zero,which is equivalent to the determinant of the matrix formed by their components being zero: $\begin{vmatrix} a & b & c \ a^2 & b^2 & c^2 \ a^3 & b^3 & c^3 \end{vmatrix} = 0$ Taking $a, b, c$ common from the first,second,and third columns respectively: $abc \begin{vmatrix} 1 & 1 & 1 \ a & b & c \ a^2 & b^2 & c^2 \end{vmatrix} = 0$ The determinant is a standard Vandermonde determinant: $abc(a-b)(b-c)(c-a) = 0$ This product is zero if $a=0$ or $b=0$ or $c=0$,or if $a=b$ or $b=c$ or $c=a$. Thus,the vectors are coplanar if either one of $a, b, c$ is zero,or any two of $a, b, c$ are equal.

Consider the vectors $u = a \hat{i} + b \hat{j} + c \hat{k}$,$v = a^2 \hat{i} + b^2 \hat{j} + c^2 \hat{k}$ and $w = a^3 \hat{i} + b^3 \hat{j} + c^3 \hat{k}$. These vectors are coplanar if and only if

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