Question

Let a, b, c be distinct non-negative numbers. If the vectors $a\mathbf{i} + a\mathbf{j} + c\mathbf{k},\mathbf{i} + \mathbf{k}$ and $c\mathbf{i} + c\mathbf{j} + b\mathbf{k}$ lie in a plane, then c is

• A. The arithmetic mean of a and b
• B. The geometric mean of a and b
• C. The harmonic mean of a and b
• D. Equal to zero

Answer 1

Answer: (B)

The geometric mean of a and b

Answer 2

a & a & c \\ 1 & 0 & 1 \\ c & c & b \end{matrix} \right| = 0 \Rightarrow \left| \begin{matrix} a & 0 & c \\ 1 & - 1 & 1 \\ c & 0 & b \end{matrix} \right| = 0$$ {Applying $C_{2} \rightarrow C_{2} - C_{1}\}$ $$\Rightarrow a( - b) + c(c) = 0 \Rightarrow c^{2} = ab.$$ Hence $c$ is the geometric mean of $a$ and $b.$