Problem 1. Answer, with justification, the following questions.

(a) What object in 3-space is traced by the tips of all vectors of the form:

(a.1) ~i +~j + c ~k, where c is any real number?

(a.2) ~i + b~j + c ~k, where b and c are any real numbers?

(b) What would you obtain if you drew all possible unit vectors in 3-space with tails at the origin?

(c) Describe the object created by all scalar multiples of ~v = ~i + ~j with tails at the point (0, 0, 1). Provide as much information as possible.

Problem 2. Given two nonzero vectors ~v and ~w, determine whether it is possible to compute the following quantities or not. If it possible to compute the given quantity, then describe it and be as complete, detailed, and precise as possible. If it is not possible to compute the given quantity, then explain why.

(a) ||~v|| ~v

(b) ~v ||~v||

(c) ||~v|| ~w

(d) ||~v|| ~w + ~w 2

(e) ||~v|| ~w + || ~w||

Problem 3.

(a) Find, with justification, all vectors ~v in the plane such that ||~v|| = 1 and ||~v +~i|| = 1.

(b) Let ~u = a~i ~j + ~k and ~v = a~i + a~j 2 ~k. For what value(s) of a R, if any, are ~u and ~v perpendicular? For what value(s) of a R, if any, are ~u and ~v parallel? Explain your reasoning.

(c) A plane has equation z = 5x 2y + 7. Explain your reasoning for all your answers.

(c.1) Find a point on the x-axis on this plane.

(c.2) Give a value of making the vector ~i +~j + 0.5 ~k normal to the plane.

(c.3) Give all possible value(s) of a so that the point P = (a + 1, a, a 1) lies on the plane.

(c.4) Find a vector parallel to the plane.

(d) For what value(s) of a, if any, are the three vectors ~a = h2, 1, 1i, ~b = h1, 2, 2i, and ~c = h3, a, 5i coplanar (i.e., lie in the same plane)? Explain your approach to this problem.

Problem 4. Suppose that ~a, ~b, and ~c are three-dimensional nonzero vectors. You have the following tasks.

Determine which of the following expressions, if any, make sense. (Assume that the symbol always means dot product, not scalar multiplication nor multiplication of numbers.)

For each expression that does make sense,

(1) decide whether the value of the expression is a scalar or a vector, and

(2) decide whether the value of the expression is zero or nonzero in each of the following situations:

(i) ~a, ~b, and ~c all point in the same direction;

(ii) ~a, ~b, and ~c all lie in the same plane, but no two are parallel;

(iii) ~a, ~b, and ~c are all perpendicular to one another.

(a) (~a ~b) ~c

(b) (~a ~b) ~c

(c) (~a ~b) ~c

(d) ~a ~b ~a ~c

(e) ~a ~b ~a ~c 2

Problem 5. Let P = (3, 4, 1), Q = (2, 0, 5), and R = (1, 2, 6). Answer, with justifications, the questions below.

(a) Find the vectors P Q, P R, and QR.

(b) Of the three vectors you found in part (a), one should be the sum of the other two. Which one is the sum? Explain your approach.

(c) Find a point S in space such that P QRS is a parallelogram; explain your approach. (Note that the name P QRS implies that the vertices of the parallelogram are P, Q, R, and S in that order.)

(d) Find the lengths of the sides of the parallelogram P QRS you found in part (c).

(e) Find the degree measures of the angles P QR and QRS to two decimal places. Explain two different approaches to find the degree measure of one of the angles.

(f) Find an equation of the plane containing the parallelogram P QRS. Include an explanation of how you derived your equation.

(g) Find the area of the parallelogram P QRS