Find the equation of the plane containing the point P = (6, 8, -12) that is parallel to the plane 6x - 4y + 10z = -20.
A triangle is formed by the points P = (1, 0, 3), Q = (4, -1, 3), and R = (6, 1, -2). (a) Sketch the triangle. (b) Find the vector and scalar equations of the plane containing this triangle.
Recall points P, Q, and R from Question 2. Suppose I have a fourth point S = (5, 3, 2). Using vectors v = PQ and w = PR, determine: (a) The area of the parallelogram formed by vectors v an w. (b) The volume of the parallelepiped formed by incorporating the point S.
Consider a parallelogram generated by the vectors <1, y, -2> and <3, 0, 1>. For what value(s) of y will the parallelogram have an areas of 8 square units?
Determine the angle between each of the following pairs of planes. (a) 3x + 2y - 7z = 4 and <2, 4, -5>
Where does the line r(t) = <1, 4, 2> + t<-1, -3, 5> intersect the plane 3x - 5y + 9z = 2, if at all?
Consider the curve r(t) =
Identify the following quadric surfaces, and sketch a graph using at least three traces for each surface. Do not use technology to graph these. (a) z = 4 - x^2 - y^2 (b) x^2 + y^2 - z^2 = -3 (c) y^2 + 9z^2 = 36 (d) x^2 + 9y^2 + 4z^2 = 36
Given the quadric surface z = 3x^2 - 2y^2 answer each of the following questions. (a) What type of quadric surface is it? How do you know? Provide the evidence supporting how you classified the quadric surface. (b) Determine the intersection points of the surface with the line r(t) = <3t, 2t, 19t>.
When a charged particle moves with velocity v through a magnetic field B, a force due to the magnetic field FB acts on the charged particle. This occurs according to the cross-product: FB = qv
Nonlinear Systems: The following nonlinear system is a model for the electric field between two point charges. Determine the equilibrium solutions, if any, and classify their stability. Then sketch the phase portrait by sketching h- and v-nullclines, drawing appropriate arrows between nullclines to determine direction of solution curves. x' = 2xy y' = y^2 - x^2 - 1
SIR model: A communicable disease is spread throughout a small community, with a fixed population of n people, by contact between infected individuals and people who are susceptible to the disease. Suppose that everyone is initially susceptible to the disease and that no one leaves the community while the epidemic is spreading. At time t, let
Symbiotic Relationship: When two species, such as the rhinoceros and birds, coexist in a symbiotic (dependent) relationship, they either increase together or decrease together. Typical equations for the growth rates of two such species might be dx1/dt = -4x1 + 4x1x2 dx2/dt = -3x2 + 2x1x2. (a) Find an equation relating x1 and x2 if x1 = 5 when x2 = 1. Meaning, write dx1/dx2 and use the separation of variables to solve the differential equation. (b) Find the equilibrium point. (c) Give a phase plane diagram (keep in mind this is a nonlinear system). (d) Based on part (c), what happens to the populations if both populations are greater than their values at equilibrium? Or, if both populations are less than their values at equilibrium?