Math 126B

Worksheet #5

Spring 2003

Names:

This worksheet looks at material from Section 12.3, most specifically the relationship:

$\mathit{a}\mathrm{ċ}\mathit{b}\mathrm{=}\mathrm{\left|}\mathit{a}\mathrm{\right|}\mathrm{\left|}\mathit{b}\mathrm{\right|}\cos \mathit{\theta }$.

1. For the following pairs of vectors, determine the value of $\mathit{a}\mathrm{ċ}$ $\mathit{b}$ and the value of $\mathit{\theta }$ (degrees).

Then, determine the relationship between the value of $\mathit{a}\mathrm{ċ}$ $\mathit{b}$ and the size of the angle between

the vectors, $\mathit{\theta }$.

The relationship between $\mathit{a}\mathrm{ċ}$ $\mathit{b}$ and the size of $\mathit{\theta }$ is...

2. Note: for a 3-dimensional graph, octants are similar to quadrants for a 2-dimensional graph.

(a) If two vectors are in the same octant, what can be said about the angle between them?

(b) How is it possible for two vectors to be in different octants and $\mathit{a}\mathrm{ċ}$ $\mathit{b}\mathrm{>}\mathrm{0}$?

Math 126B

Worksheet #5

Spring 2003

3. Let's look at the situation where $\mathit{a}\mathrm{ċ}$ $\mathit{b}\mathrm{=}\mathrm{0}$.

(a) What is the geometric relationship between two vectors when $\mathit{a}\mathrm{ċ}$ $\mathit{b}\mathrm{=}\mathrm{0}$? (In other words,

what is the angle between two vectors when their dot product is 0?)

(b) Find two different vectors that are perpendicular (orthogonal) to both of the vectors

$\mathrm{⟨}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{0}\mathrm{⟩}$ and $\mathrm{⟨}\mathrm{1}\mathrm{,}\mathrm{}\mathrm{0}\mathrm{,}\mathrm{}\mathrm{1}\mathrm{⟩}$

4. Suppose that $\mathit{a}$ and $\mathit{b}$ are nonzero vectors.

(a) Dude! If $\mathit{a}\mathrm{ċ}$ $\mathit{b}\mathrm{=}\mathit{b}\mathrm{ċ}$ $\mathit{a}$, then how is it possible that $\mathrm{c}\mathrm{o}\mathrm{m}{\mathrm{p}}_{\mathit{a}}\mathit{b}$ and $\mathrm{c}\mathrm{o}\mathrm{m}{\mathrm{p}}_{\mathit{b}}\mathit{a}$ are different?

(b) What is the special relationship between the vector $\mathit{b}\mathrm{-}\mathrm{p}\mathrm{r}\mathrm{o}{\mathrm{j}}_{\mathit{a}}\mathit{b}$ and the vector $\mathit{a}$? Give

an example to illustrate your observation.