1. Given:
Three points A(1, 0, 0) and B(0, 2, 0) and C(0, 0, 3) which can be visualized as defining the triangle ABC located in the first octant of a 3D Cartesian coordinate system.
Find:
a. A unit vector u normal to plane ABC, visualized as extending from the origin into the first octant.
b. A position vector r in the direction of u whose tip extends to a point D on plane ABC.
c. The (x, y, z) coordinates of this point D.
d. The coefficients, a, b, and C of the line Z = ax + by + C passing through points A and D.
e. Consider a particle whose path is described by the position vector r, measured in meters, where r is a linear function of time, measured in seconds. If it take one second for this particle to travel from the origin to point D, how long would it take for the particle to travel 10 meters? What would it's position be at that time.
f. A 3D Cartesian coordinate system sketch, roughly to scale, showing the:
i. Points A, B, C and D with a triangle connecting points A, B and C.
ii. Vectors u, r and line through points A and D.
Hints: I recommend that you start by working with a rough draft of the sketch required in part e to help you visualize how to compute the various results required in the problem. You may want to use dotted lines to indicate any vectors located "behind" plane ABC. Recall the several characteristics of the dot and cross product operations we discussed in class. These should be helpful to you in solving this problem.

2. Given:
The space curve traced out by r(t) = < sin(t), cos(3t) >
Find:
a. a plot of the path described by the parametric vector function r(t) in the x-y plane over the interval (0, π]
b. Expressed as a definite integral, the length of the space curve segment described by r(t) over the interval (0, π]
c. The unit tangent vector T, as a function of the parameter t.
d. The scalar magnitude of the curvature K as a function of the parameter t (no need to try and simplify this expression).
Hints: Don't worry about simplifying your results. Specifically, you can leave the curve length expressed as a definite integral in part b and you don't need to try and use trig identities to simplify the unit tangent vector curvature expression you derive in part d.

1. Given: Three points A(1, 0, 0) and B(0, 2, 0) and C(0, 0, 3) wh...