Please help me to solve these problems and write answers down and also explain r
ID: 3145552 • Letter: P
Question
Please help me to solve these problems and write answers down and also explain reasons. Thank you so much. In this problem, answer "True" or "False" for each question. Note: there is no partial credit for this problem. You must answer all parts correctly to receive credit. You will not be shown the correct answers for individual parts 1. If two lines in R do not intersect, they must be parallel True OFalse 2. A line in R3 is completely determined by two different points on it. OTrue False 3. A plane in IR3 is completely determined by a point in the plane and any non-zero vector parallel to the plane True O False 4·Three different planes in R3 intersect in a point or not at all True FalseExplanation / Answer
1) False. In 3 dimensions, you can imagine two lines that don't intersect, but are not parallel. For example, take a road with an overpass. The road goes one way (north-south). The overpass goes another way (east-west). They never intersect, but they aren't parallel. Such lines are called skew lines.
2) True. A line is completely determined by any two distinct points that lie on it. However, this is the same as saying that a line is completely determined by a point on it, and a direction vector for that line, where a direction vector is any vector whose endpoints are two distinct points on the line.If (x0, y0, z0) is any point on a line l, and a, b, c a direction vector for l, then the line is given in parametric form by
x(t) = x0+at, y(t) = y0+bt, z(t) = z0+ct
3) False. A plane in the 3D coordinate space is determined by a point and a vector that is perpendicular to the plane.
4) False. In 3D, three planes , P1, P2, and P3 can intersect (or not) in the following ways:
(a) all three planes are parallel.
(b) just two planes are parrallel and the third plane cuts each in a line.
(c) the intersection of the three planes is a line.
(d) the intersection of the three planes is a point.
(e) each plane cuts the other two in a line.
(f) two coincident planes and the other intersecting them in a line.