Title: Math 2260 Chapter 11 (Vectors and Geometry of 3-space)

Q: [latex]
\begin{large}
The distance between $(x_1, y_1, z_1)$ and $(x_2,y_2,z_2)$
\end{large}
[/latex]
A: [latex]
\begin{equation*} \sqrt{ (x_1 - x_2)^2 + (y_1 - y_2)^2 + (z_1 - z_2)^2} \end{equation*}
[/latex]

Q: [latex]
\begin{large}
The equation of a sphere of radius $r$ centered at $(x_0,y_0,z_0)$:
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*} (x - x_0)^2 + (y - y_0)^2 + (z - z_0)^2 = r^2.
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The length of the vector $\vec{v} = (v_1, v_2, v_3)$ is
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*} |\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2} \end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
If $\vec{u} = (u_1,u_2,u_3)$ and $\vec{v} = (v_1,v_2,v_3)$ then $\vec{u} + \vec{v} = $
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*} \vec{u} + \vec{v} = (u_1 + v_1, u_2 + v_2, u_3 + v_3). \end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The product of a vector $\vec{u} = (u_1,u_2,u_3)$ and a scalar $k$ is $k \vec{u} = $
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*}
k \vec{u} = (k u_1, k u_2, k u_3).
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The unit vector in direction $\vec{v}$ is
\end{large}
[/latex]
A: [latex]
\begin{large}
the scalar product $(1/|\vec{v}|) \vec{v}$.
\end{large}
[/latex]

Q: [latex]
\begin{large}
The dot product of two vectors $\vec{u}$ and $\vec{v}$, $\vec{u} \cdot \vec{v}$ or $\left< \vec{u}, \vec{v} \right>$ is given by
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*} \left< \vec{u}, \vec{v} \right> = u_1 v_1 + u_2 v_2 + u_3 v_3.
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The relationship between the \emph{dot product} $\left< \vec{u}, \vec{v} \right>$ and the \emph{angle} $\theta$ between vectors $\vec{u}$ and $\vec{v}$ is
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*} \left< \vec{u}, \vec{v} \right> = |\vec{u}| |\vec{v}| \cos \theta.
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
Vectors $\vec{u}$ and $\vec{v}$ are \emph{orthogonal} if
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*} \left< \vec{u}, \vec{v} \right> = 0.
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The \emph{cross product} of two vectors $\vec{u}$ and $\vec{v}$ in $3$-space is given by
\begin{equation*}
\vec{u} \times \vec{v} =
\end{equation*}
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*} \vec{u} \times \vec{v} = ( u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1 ).
\end{equation*}
These are the terms in the ``determinant''
\begin{equation*}
\vec{u} \times \vec{v} =
\begin{vmatrix}
\vec{e_1} & \vec{e_2} & \vec{e_3} \\
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3
\end{vmatrix}
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The relationship between the \emph{cross product} of vectors $\vec{u}$ and $\vec{v}$ and the \emph{angle} $\theta$ between $\vec{u}$ and $\vec{v}$ is
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*}
\vec{u} \times \vec{v} = |\vec{u}| \, |\vec{v}| \sin \theta.
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The relationship between $\vec{u} \times \vec{v}$ and $\vec{v} \times \vec{u}$ is
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*} \vec{u} \times \vec{v} = - \vec{v} \times \vec{u}.
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The relationship between \emph{cross product} and \emph{area} is
\end{large}
[/latex]
A: [latex]
\begin{large} $\vec{u} \times \vec{v}$ is the area of the parallelogram with sides $\vec{u}$ and $\vec{v}$.
\end{large}
[/latex]

Q: [latex]
\begin{large}
The volume of the parallelepiped (box) with sides $\vec{u}$, $\vec{v}$, and $\vec{w}$ is given by
\end{large}
[/latex]
A: [latex]
\begin{large}
The \emph{triple product} $(\vec{u} \times \vec{v}) \cdot \vec{w}$, which is also given by the determinant
\begin{equation*}
\begin{vmatrix}
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3 \\
w_1 & w_2 & w_3
\end{vmatrix}
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The (vector) equation for a line through $\vec{p}$ parallel to $\vec{v}$ is
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*}
\vec{r}(t) = \vec{p} + t \vec{v}.
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The vector equation for a plane through $\vec{p}$ with normal vector $\vec{n}$ is
\end{large}
[/latex]
A: [latex]
\begin{large}
\begin{equation*}
\vec{n} \cdot (\vec{x} - \vec{p}) = 0
\end{equation*}
or
\begin{equation*}
n_1 x_1 + n_2 x_2 + n_3 x_3 = n_1 p_1 + n_2 p_2 + n_3 p_3.
\end{equation*}
\end{large}
[/latex]

Q: [latex]
\begin{large}
The angle between two planes is given by
\end{large}
[/latex]
A: [latex]
\begin{large}
the angle between their normal vectors.
\end{large}
[/latex]