Consider the vectors $u=a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}$, $v=a^2…
Consider the vectors $u=a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}$, $v=a^2 \hat{\mathbf{i}}+b^2 \hat{\mathbf{j}}+c^2 \hat{\mathbf{k}}$ and $w=a^3 \hat{\mathbf{i}}+b^3 \hat{\mathbf{j}}+c^3 \hat{\mathbf{k}}$.
These vectors are coplanar if and only if
all $a, b$ and $c$ are equal
one of $a, b$ and $c$ is zero
any two of $a, b$ and $c$ are equal
either one of $a, b$ and $c$ is zero, or any two of $a, b$ and $c$ are equal
Solution
Here
$
\begin{aligned}
\mathbf{u} & =a \hat{\mathbf{i}}+b \hat{\mathbf{J}}+c \hat{\mathbf{k}} \\
\mathbf{v} & =a^2 \hat{\mathbf{i}}+b^2 \hat{\mathbf{j}}+c^2 \hat{\mathbf{k}} \\
\mathbf{w} & =a^3 \hat{\mathbf{i}}+b^3 \hat{\mathbf{j}}+c^3 \hat{\mathbf{k}}
\end{aligned}
$
$\mathbf{u}, \mathbf{v}, \mathbf{w}$ are coplanar.
$
\begin{aligned}
& \therefore\left|\begin{array}{lll}
a & b & c \\
a^2 & b^2 & c^2 \\
a^3 & b^3 & c^3
\end{array}\right|=0 \\
& \Rightarrow a b c\left|\begin{array}{ccc}
1 & 1 & 1 \\
a & b & c \\
a^2 & b^2 & c^2
\end{array}\right|=0 \\
& \Rightarrow a b c\left|\begin{array}{ccc}
1 & 0 & 0 \\
a & b-c & c-a \\
a^2 & b^2-c^2 & c^2-a^2
\end{array}\right|=0 \text { [using } \\
& \Rightarrow a b c(a-b)(b-c)(c-a)=0
\end{aligned}
$
$\therefore$ Either one of $a, b, c$ is zero or any two of $a, b, c$ are equal