Name: ___________________

Math 122 Lab 5: Vectors and Geometry

Introduction

One of the most powerful aspects of vectors is the strong connection to geometry. Often, formulas that are very complicated algebraically are immediately obvious using vector geometry. In this assignment, you will explore some of the relationships between vectors and geometry.

What to Turn In

Turn in this worksheet with everything filled in. This will be graded for accuracy and understanding, so write good explanations when you are asked to explain something. If you need more space, turn in additional pages typed or neatly written. Late papers lose 4 points per day late.

Three-Dimensional Geometry

Use vectors to determine if the points (0,1,1), (2,4,2) and (3,1,4) form an equilateral triangle. Briefly explain why your method works.

Use vectors to determine if the points (3,1,-2), (1,0,1) and (4,2,-1) form a right triangle. Briefly explain why your method works.

The Triangle Inequality

The triangle inequality states that for any vectors a and b, ||a + b|| ≤ ||a|| + ||b||. Draw a picture of vectors a, b and a + b and use it to explain why the inequality is obvious. (Hint: use the phrase “The shortest path between …”)

Based on the geometry of the above picture, under what circumstances would you get equality; that is, when is ||a + b|| = ||a|| + ||b|| ?

Use geometry to determine the circumstances in which ||a + b||² = ||a||² + ||b||².

Geometric Explorations.

(1) In general, ||a + b||² ≠ ||a||² + ||b||². Use examples to determine which side is larger. (Hint; there is more than one case.)

(2) Describe the set of all points in three dimensions that are equidistant from two fixed points A and B. Sketch a picture. For points A = (0,0,0) and B = (1,2,4) find an equation of this object.