MHT CET · Maths · Vector Algebra
Let \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) be distinct non-negative umbers. If 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 a plane, then c is
- A the arithmetic mean of a and b.
- B the harmonic mean of a and \(b\).
- C the geometric mean of \(a\) and \(b\).
- D not arithmetic mean of a and b.
Answer & Solution
Correct Answer
(C) the geometric mean of \(a\) and \(b\).
Step-by-step Solution
Detailed explanation
\(\begin{aligned} & \left|\begin{array}{lll}a & a & c \\ 1 & 0 & 1 \\ c & c & b\end{array}\right|=0 \\ & \Rightarrow a b-c^2=0 \\ & \Rightarrow c^2=a b\end{aligned}\)
\(\Rightarrow\) The geometric mean of a and b is \(\mathrm{c}\)
\(\Rightarrow\) The geometric mean of a and b is \(\mathrm{c}\)
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