![[Maple Plot]](Quiz11.gif)
(e) Showing all your work, calculate the distance from the line L(t) found in (c), to the point p in the plane found in (a).
1. Let n = < 0, 0, 1>, r
= <x, y, z>, p = < 1, 1, 0>, v = < -1, 1, 0>, and
q = < 1, 1, 1>.
(a) Write down the vector equation of
the plane determined by n, r, and p, where n is the
normal to the plane, and p is a point lying in the plane.
(b) Simplify
the vector equation of the plane found in (a). What plane is it?
(c) Write
down the vector equation of the line L(t) determined by v
and q, where v is the direction vector of the the line, and
q is a point lying on the line.
(d) Carefully sketch
the vector equation of the plane in (a), and the the vector equation
of the line L(t) in (c), on the plot below:
(e) Showing all your work,
calculate the distance from the line L(t) found in (c), to
the point p in the plane found in (a).
Name _______________________
Calculus III, Quiz 2
Page 2
2. Each of the following two planes
is determined by a normal vector and a point:
Plane1: n1 = < 0, -1, 0>, p1
= < 1, 0, -1>, and Plane2: n2 = < -1, 1, 0>, and p2
= < 1, 1, 1>.
(a) Carefully sketch Plane1and
Plane2 below:
(b) Showing all your work find
the acute angle made by the intersection of Plane1and Plane2:
(c) Showing all your work find
the distance from Plane1 to p2.
(d) Showing all your work find
the distance from Plane2 to p1.
Name _______________________
Calculus III, Quiz 2
Page 3
3. Let v1 = < 1, 1, 1>,
p1 = < 0, 0, 0>, v2 = < 1, 2, 3>, p2 = <
-1, 0, 0>, and define two lines as follows:
Line1: L1(t) = v1t + p1 and Line2: L2(s)
= v2s + p2.
(a) Find the
vector w = L2(s) - L1(t).
(b) Solve the system of two equations in two unknowns (s and t), given by the following two vector equations:
w dot v1 = 0
w dot v2 = 0
(c) Next,
find | w | using the values for s and t found in (b).
(d) Finally, what is the geometrical significance of | w | in relation to Line1 and Line2? Explain.