**Previous section: ****Rules for Multiplication by Scalars****.**

**Next section: ****Introduction to Geometric Algebra and Basic 2D Applications****.**

**Exercise: **As appropriate, identify and sketch the indicated geometric figures below.

When is it necessary to designate a particular point by the zero vector?

• **Line: x**(α) = α**a **+** b**

** Line segment **for 0 ≤ α ≤1

• **Plane: x**(α,β) = α**a **+** **β**b **+** c**

** Linear constraints: **Sketch lines for α = 1, 2, 3 and then for β = 1, 2, 3.

(Click **here** for a solution sketch.)

** Quadratic constraints: **(Choose the most symmetrical parametric form)

1. α^{2 }+ β^{2 }= 1

2. β = α^{2}

3. α^{2 }− β^{2 }= 1

(Click **here** for solution sketches.)

**News Release: **Physics Education Research in a large state university found that, after completing a semester of introductory physics, most students were unable to carry out graphical vector addition in two dimensions. The more complex skills of coordinating scalar multiplication with vector addition were not investigated. [Nguyen & Meltzer, *AJP ***71**: 630-638 (2003)]

**Question: **What are likely reasons for this unacceptable failure of mathematics instruction?

Answers:

- Failure of the math curriculum to provide timely instruction in vector methods.
- Overreliance on coordinate methods in most courses.
- Vectors are only sporadically employed and usually with orthogonal bases, so students have little opportunity to develop fluency with the general features of vector algebra listed above.
- Students are unclear about the geometric interpretation of vectors (see below)
(see the sections on GA, which fills this gap).**Vector algebra is incomplete without rules for multiplying vectors that encode information about magnitudes and relative direction**

**Previous section: ****Rules for Multiplication by Scalars****.**

**Next section: ****Introduction to Geometric Algebra and Basic 2D Applications****.**