If a, b, c are distinct non-negative numbers | vedclass

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(B) Three vectors are coplanar if their scalar triple product is zero.Let the vectors be $\vec{u} = a\hat{i} + a\hat{j} + c\hat{k}$, $\vec{v} = \hat{i

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The geometric mean of $a$ and $b$

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(B) Three vectors are coplanar if their scalar triple product is zero.Let the vectors be $\vec{u} = a\hat{i} + a\hat{j} + c\hat{k}$, $\vec{v} = \hat{i} + 0\hat{j} + \hat{k}$, and $\vec{w} = c\hat{i} + c\hat{j} + b\hat{k}$.The condition for coplanarity is $\det(\vec{u}, \vec{v}, \vec{w}) = 0$.$\begin{vmatrix} a & a & c \ 1 & 0 & 1 \ c & c & b \end{vmatrix} = 0$.Expanding along the second row: $-1(ab - c^2) + 0 - 1(ac - ac) = 0$.$-(ab - c^2) = 0$.$c^2 = ab$.$c = \sqrt{ab}$.Thus, $c$ is the geometric mean of $a$ and $b$.

If $a, b, c$ are distinct non-negative numbers and the vectors $a\hat{i} + a\hat{j} + c\hat{k}$, $\hat{i} + \hat{k}$, and $c\hat{i} + c\hat{j} + b\hat{k}$ lie in the same plane, then the value of $c$ is...

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