Calculus- Early Transcendentals, 2021a

12.5. Lines and Planes 443
Exercise 12.5.3 Find an equation of the plane containing (1,2,−3), (0,1,−2) and (1,2,−2).
Exercise 12.5.4 Find an equation of the plane containing (1,0,0), (4,2,0) and (3,2,1).
Exercise 12.5.5 Find an equation of the plane containing (1,0,0) and the line 〈1,0,2〉 +t〈3,2,1〉.
Exercise 12.5.6 Find an equation of the plane containing the line of intersection of x + y + z = 1 and
x − y + 2z = 2, and perpendicular to the xy-plane.
Exercise 12.5.7 Find an equation of the line through (1,0,3) and (1,2,4).
Exercise 12.5.8 Find an equation of the line through (1,0,3) and perpendicular to the plane x + 2y − z =
1.
Exercise 12.5.9 Find an equation of the line through the origin and perpendicular to the plane x+y−z =
2.
Exercise 12.5.10 Find a and c so that (a,1,c) is on the line through (0,2,3) and (2,7,5).
Exercise 12.5.11 Explain how to discover the solution in Example 12.12.
Exercise 12.5.12 Determine whether the lines 〈1,3,−1〉 + t〈1,1,0〉 and 〈0,0,0〉 + t〈1,4,5〉 are parallel,
intersect, or neither.
Exercise 12.5.13 Determine whether the lines 〈1,0,2〉 +t〈−1,−1,2〉 and 〈4,4,2〉 +t〈2,2,−4〉 are parallel,
intersect, or neither.
Exercise 12.5.14 Determine whether the lines 〈1,2,−1〉 + t〈1,2,3〉 and 〈1,0,1〉 + t〈2/3,2,4/3〉 are parallel,
intersect, or neither.
Exercise 12.5.15 Determine whether the lines 〈1,1,2〉 +t〈1,2,−3〉 and 〈2,3,−1〉 +t〈2,4,−6〉 are parallel,
intersect, or neither.
Exercise 12.5.16 Find a unit normal vector to each of the coordinate planes.
Exercise 12.5.17 Show that 〈2,1,3〉 +t〈1,1,2〉 and 〈3,2,5〉 + s〈2,2,4〉 are the same line.
Exercise 12.5.18 Give a prose description for each of the following processes:
(a) Given two distinct points, find the line that goes through them.
(b) Given three points (not all on the same line), find the plane that goes through them. Why do we need
the caveat that not all points be on the same line?
(c) Given a line and a point not on the line, find the plane that contains them both.
(d) Given a plane and a point not on the plane, find the line that is perpendicular to the plane through
the given point.