Question

is the following statement true or false? the intersection of three planes can be a point. true false

Explanation:

Step1: Recall plane intersection concepts

In 3 - D space, consider three planes. For example, the three coordinate planes (xy - plane, yz - plane, xz - plane) intersect at the origin \((0,0,0)\), which is a single point. In general, if three planes are not parallel and their pairwise intersection lines are not parallel to each other, they can intersect at a single point. Mathematically, we can represent planes as \(a_1x + b_1y + c_1z=d_1\), \(a_2x + b_2y + c_2z=d_2\), \(a_3x + b_3y + c_3z=d_3\). The system of linear equations formed by these planes can have a unique solution (a point) if the determinant of the coefficient matrix is non - zero (in the case of non - parallel, non - coincident planes with non - parallel intersection lines).

Step2: Determine the truth value

Since there are cases (like the coordinate planes) where three planes intersect at a single point, the statement "The intersection of three planes can be a point" is true.

Answer:

true