Question

Let a, b, c be distinct non-negative numbers. 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 Geometric Mean of a and b
• C. The Harmonic Mean of a and b
• D. Equal to zero

Hint:

The condition for three vectors to be coplanar is that their scalar triple product must be zero. This is most easily calculated by setting the determinant of the vectors' components to zero and solving.

Answer 1

The Correct Option is B

If three vectors are coplanar (lie in the same plane), their scalar triple product is zero. The scalar triple product is calculated as the determinant of the matrix formed by the components of the vectors. 

Let the vectors be \( \vec{u} = a\hat{i} + a\hat{j} + c\hat{k} \), \( \vec{v} = 1\hat{i} + 0\hat{j} + 1\hat{k} \), and \( \vec{w} = c\hat{i} + c\hat{j} + b\hat{k} \). 

For them to be coplanar, \( [\vec{u} \vec{v} \vec{w}] = 0 \). \[ \begin{vmatrix} a & a & c \\ 1 & 0 & 1 \\ c & c & b \end{vmatrix} = 0 \] Expand the determinant along the second row for easier calculation: \\ \[ -1(ab - c^2) + 0 - 1(ac - ac) = 0 \] \[ -ab + c^2 = 0 \] \[ c^2 = ab \] Since a, b, c are non-negative numbers, we can take the square root: \[ c = \sqrt{ab} \] 

This is the definition of the Geometric Mean of a and b.