MATH 2310 Homework 2 Spring 2022
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MATH 2310
Homework 2
Spring 2022
1. [4 pts] Let a and b be two non-zero vectors in R3. Prove the following expression ∥a × b∥2 = ∥a∥2 · ∥b∥2 − (a · b)2
2. [4 pts] Suppose that for two three-dimensional vectors a and b it is the case that a × c = b × c for every three-dimensional vector c. Prove that this requires that a = b, or provide an example or argument that disproves it.
3. Consider the points P(3, 1, 1),Q(4, 1, 2), and R(4, 4, 1) in R3 .
(a) [3 pts] Find an equation for the plane containing the points P,Q, and R.
(b) [2 pts] Find the area of the triangle with vertices P,Q, and R.
4. [4 pts] Find an equation of the plane which passes through the points (2 , 2, 1) and ( − 1, 1, − 1) and is perpendicular to the plane 2x − 3y + z = 3.
5. [4 pts] Find a set of parametric equations for the line of intersection of the planes:
6x − 3y + z = 5
−x +y + 5z = 5
2022-02-10