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

Correct Answer is : m,n,p{m}, {n}, {p}m,n,p are in G.P.

Correct Answer is : m,n,p{m}, {n}, {p}m,n,p are in G.P.

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MHTCET 22 Apr 2025 Shift 1 Paper

Question : 124 of 150

Marks: +1, -0

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...

Solution:

Concept:

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

Explanation:

Let the three vectors be:

\vec{A}=m\hat{i}+m\hat{j}+n\hat{k}

\vec{B}=\hat{i}+\hat{k}

\vec{C}=n\hat{i}+n\hat{j}+p\hat{k}

Since they lie in a plane,

\vec{A}\cdot(\vec{B}\times\vec{C})=0

This gives the determinant:

\begin{vmatrix} m & m & n\\ 1 & 0 & 1\\ n & n & p \end{vmatrix}=0.

Expanding along the first row:

m(0\cdot p-1\cdot n)-m(1\cdot p-1\cdot n)+n(1\cdot n-0\cdot n)=0

\Rightarrow m(-n)-m(p-n)+n(n)=0

\Rightarrow -mn-mp+mn+n^2=0

\Rightarrow n^2-mp=0

\Rightarrow n^2=mp

Thus, the middle term is the geometric mean of the other two.

m

n

p

are in G.P.

Answer:

Option C —

m

n

p

are in G.P.

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