Homework 9 - Physics 265

Physics 265 Physics 265
Introductory Scientific Computing
Assignment # 10: Vectors and Arrays

Dealing with vectors that are represented by components is really quite easy. For example, let's say that we have a vector

A

that we represent with its three components,

A = [\begin{matrix} A_{x} \\ A_{y} \\ A_{z} \end{matrix}], (5)

as well as another vector

B

that we represent by its three components,

B = [\begin{matrix} B_{x} \\ B_{y} \\ B_{z} \end{matrix}] . (6)

Then all possible vector operations can be defined in terms of the components of $A$ and $B$ . For example,

\begin{matrix} A + B & = & [\begin{matrix} A_{x} + B_{x} \\ A_{y} + B_{y} \\ A_{z} + B_{z} \end{matrix}], & (7) \\ A - B & = & [\begin{matrix} A_{x} - B_{x} \\ A_{y} - B_{y} \\ A_{z} - B_{z} \end{matrix}], & (8) \\ c A & = & [\begin{matrix} {cA}_{x} \\ {cA}_{y} \\ {cA}_{z} \end{matrix}], & (9) \\ A \cdot B & = & A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}, & (10) \\ A \times B & = & [\begin{matrix} A_{y} B_{z} - A_{z} B_{y} \\ A_{z} B_{x} - A_{x} B_{z} \\ A_{x} B_{y} - A_{y} B_{z} \end{matrix}] . & (11) \end{matrix}

[Actually, the cross product

A \times B

is perpendicular to the plane formed by

A

and

B

, which also makes it perpendicular to both

A

and

B

Do a matrix multiplication to determine $L$ for the three cases in which
$ω = [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] . (29)$
For the $L$ and last value of $ω$ , determine the cross product $ω \times L$ and the dot product $ω \cdot ω$ .