Question

part 2 of 3
(b) the intersection of two distinct lines is a point.
part 3 of 3
(c) the intersection of three distinct planes is a line.

Explanation:

Step1: Analyze the intersection of three distinct planes

Three distinct planes can intersect in different ways.

• Case 1: If three planes are parallel to each other, they have no intersection.
• Case 2: If two planes are parallel and the third intersects them, there is no common intersection for all three planes.
• Case 3: Three planes can intersect at a single point. For example, consider the three coordinate planes \(x = 0\), \(y=0\), \(z = 0\) in 3 - D space. Their intersection is the origin \((0,0,0)\).
• Case 4: Three planes can intersect along a line. For example, consider the planes \(x+y+z = 1\), \(2x + 2y+2z=2\) (which is just a scalar multiple of the first plane, but if we take non - scalar - multiple planes like \(x = 0\), \(y+z=0\), \(y - z = 0\). The intersection of \(y + z=0\) and \(y - z=0\) gives \(y=0,z = 0\), and when combined with \(x = 0\) (the \(yz\) - plane), the intersection is the \(x\) - axis (\(y = 0,z = 0\) for all \(x\))

Answer:

Sometimes true