Question

If the vectors \(m\hat{i} + m\hat{j} + n\hat{k}, \hat{i} + \hat{k}, n\hat{i} + n\hat{j} + p\hat{k}\) lie in a plane then...

• A. \(m + n + p = 0\)
• B. \(m, n, p\) are in A.P.
• C. \(m, n, p\) are in G.P.
• D. \(n, m, p\) are in G.P.

Hint:

For three vectors to be coplanar, always use: \[ [\vec{a}\ \vec{b}\ \vec{c}] = 0 \] That usually reduces quickly to a simple relation among constants.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Coplanar vectors have a scalar triple product of zero.
Step 2: Key Formula or Approach:
Determinant \(\begin{vmatrix} m & m & n
1 & 0 & 1
n & n & p \end{vmatrix} = 0\).
Step 3: Detailed Explanation:
Expand: \(m(0 - n) - m(p - n) + n(n - 0) = 0\)
\(-mn - mp + mn + n^2 = 0 \implies n^2 = mp\).
Step 4: Final Answer:
\(m, n, p\) are in G.P.