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

Concept:
Three vectors are coplanar if their scalar triple product is zero. So we form the determinant using the components of the three vectors and set it equal to zero. ip

Step 1: Write the three vectors in component form.
\[ \vec{a}=(m,m,n), \qquad \vec{b}=(1,0,1), \qquad \vec{c}=(n,n,p) \] ip

Step 2: Use the coplanarity condition.
For coplanarity, \[ \begin{vmatrix} m & m & n \\ 1 & 0 & 1 \\ n & n & p \end{vmatrix} =0 \] ip

Step 3: Expand the determinant.
Expanding, \[ m(0\cdot p-1\cdot n)-m(1\cdot p-1\cdot n)+n(1\cdot n-0\cdot n)=0 \] \[ m(-n)-m(p-n)+n^2=0 \] \[ -mn-mp+mn+n^2=0 \] \[ n^2-mp=0 \] \[ n^2=mp \] ip

Step 4: Interpret the relation.
The condition \[ n^2=mp \] means that \(m,n,p\) are in geometric progression. ip Hence, the correct answer is:
\[ \boxed{(C)\ m,n,p \text{ are in G.P.}} \]