Question

The equations \[ a_1x + b_1y + c_1z = d_1,\quad a_2x + b_2y + c_2z = d_2,\quad a_3x + b_3y + c_3z = d_3 \] represent three planes. Given that \( d_1d_2d_3 \ne 0 \) and \[ \frac{a_i}{a_j} \ne \frac{b_i}{b_j} \ne \frac{c_i}{c_j} \quad \text{for } i, j = 1, 2, 3,\ i \ne j, \] Let \[ A = \begin{bmatrix} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \end{bmatrix}, \quad B = \begin{bmatrix} d_1 \\ d_2 \\ d_3 \end{bmatrix}. \] If \( \text{rank}(A) = \text{rank}([A:B]) = 2 \), then the planes are such that

• A. line of intersection of any two is parallel to the third plane
• B. line of intersection of any two planes cuts the third plane
• C. line of intersection of any two planes lies on the third plane
• D. the three planes are adjacent faces of a box

Hint:

If \( \text{rank}(A) = \text{rank}([A:B]) = 2 \), the system is consistent and the solution represents a line.

Answer 1

The Correct Option is C

Given that the rank of matrix \( A \) and augmented matrix \( [A:B] \) is 2, it implies that the three planes intersect in a line (not a point and not all three being parallel).
If each pair of planes intersects in a line and these lines lie on the third plane, then all three planes intersect in the same line.
This configuration corresponds to the case where every two planes intersect in a line and all such lines lie in the third plane, indicating a common line of intersection.