Question

If \(a,b,c\) are distinct non-negative numbers and the vectors \(a\hat{i}+a\hat{j}+c\hat{k},\hat{i}+\hat{k},c\hat{i}+c\hat{j}+b\hat{k}\) lie in the same plane, then the value of \(c\) 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. \(0\)

Hint:

Three vectors are coplanar when their scalar triple product is zero.

Answer 1

The Correct Option is B

Step 1: Set up the determinant
\[ \begin{vmatrix}a&a&c\\1&0&1\\c&c&b\end{vmatrix}=0 \]

Step 2: Expand
\(a(0\cdot b-c)-a(b-c)+c(c-0)=-ac-ab+ac+c^2=c^2-ab\).

Step 3: Conclude
\(c^2=ab\), so \(c=\sqrt{ab}\), the geometric mean of \(a\) and \(b\). Option (B).

Final Answer:
\(c\) is the geometric mean of \(a\) and \(b\), option (B). \[ \boxed{\text{(B)}} \]