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

Q: [latex]The distance between $\vec{v} = (v_1, v_2,  \dots, v_n)$ and $\vec{w} = (w_1,w_2,\dots,w_n)$[/latex]
A: [latex]$$ |\vec{v} - \vec{w}| = \sqrt{ (v_1 - w_1)^2 + (v_2 - w_2)^2 + \cdots + (v_n - w_n)^2} $$ [/latex]

Q: [latex]The equation of a sphere of radius $r$ centered at $\vec{c} = (c_1,c_2, \dots, c_n)$[/latex]
A: [latex]$$ (x_1 - c_1)^2 + (x_2 - c_2)^2 + \cdots + (x_n - c_n)^2 = r^2 $$ [/latex]

Q: [latex]The length or magnitude of the vector $\vec{v} = (v_1, v_2, \dots, v_n)$ is[/latex]
A: [latex]$$ |\vec{v}| = \sqrt{v_1^2 + v_2^2 + \cdots +  v_n^2} $$ [/latex]

Q: [latex]If $\vec{u} = (u_1,u_2,\dots,u_n)$ and $\vec{v} = (v_1,v_2,\dots,v_n)$ then
$\vec{u} + \vec{v} = $
[/latex]
A: [latex]$$ \vec{u} + \vec{v} = (u_1 + v_1, u_2 + v_2, \dots, u_n + v_n).$$[/latex]

Q: [latex]
The product of a vector $\vec{u} = (u_1,u_2,\dots,u_n)$ and a scalar $\lambda$ is $\lambda \vec{u} = $[/latex]
A: [latex]$$ \lambda \vec{u} = (\lambda u_1, \lambda u_2, \cdots, \lambda u_n). $$[/latex]

Q: [latex]The unit vector in direction $\vec{v}$ is[/latex]
A: [latex]the scalar product $(1/|\vec{v}|) \vec{v}$ or
$$\frac{\vec{v}}{|\vec{v}|}$$
Sometimes, this is called $\hat{v}$.
[/latex]

Q: [latex]The dot product of two vectors $\vec{u}$ and $\vec{v}$ in $\mathbb{R}^n$
$\vec{u} \cdot \vec{v}$
or
$\left< \vec{u}, \vec{v} \right>$
is given by
[/latex]
A: [latex]
$$\left< \vec{u}, \vec{v} \right> = u_1 v_1 + u_2 v_2 + \dots u_n v_n.$$
[/latex]

Q: [latex]The relationship between the dot product $\left< \vec{a}, \vec{b} \right>$ and the angle $\theta$ between vectors $\vec{a}$ and $\vec{b}$ is[/latex]
A: [latex]$$\left< \vec{a}, \vec{b} \right> = |\vec{a}| |\vec{b}| \cos \theta.$$[/latex]

Q: [latex]Vectors $\vec{u}$ and $\vec{v}$ are orthogonal or perpendicular if[/latex]
A: [latex]
$$\left< \vec{u}, \vec{v} \right> = 0.$$
[/latex]

Q: [latex]
The cross product of two vectors $\vec{u}$ and $\vec{v}$ in $\mathbb{R}^3$ is given by
$$\vec{u} \times \vec{v} = $$
[/latex]
A: [latex]
$$\vec{u} \times \vec{v} = ( u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1 ).$$
These are the terms in the ``determinant''
$$\vec{u} \times \vec{v} =
\begin{vmatrix}
\vec{i} & \vec{j} & \vec{k} \\
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3
\end{vmatrix}$$
where $\vec{i} = (1,0,0)$, $\vec{j} = (0,1,0)$, and $\vec{k} = (0,0,1)$ and we ``expand by minors'' to get
$$\vec{i} (u_2 v_3 - u_3 v_2) - \vec{j} (u_1 v_3 - u_3 v_1) + \vec{k} (u_1 v_2 - u_2 v_1).$$
[/latex]

[latex]The relationship between the cross product of vectors $\vec{a}$ and $\vec{b}$ and the angle $\theta$ between $\vec{a}$ and $\vec{b}$ is [/latex]

Q: [latex]The relationship between \\
$\vec{a} \times \vec{b}$ and $\vec{b} \times \vec{a}$ is[/latex]
A: [latex]
$$\vec{a} \times \vec{b} = - \vec{b} \times \vec{a}$$
[/latex]

Q: [latex]The relationship between cross product and area is[/latex]
A: [latex]The length $|\vec{a} \times \vec{b}|$ is the area of the parallelogram with sides $\vec{a}$ and $\vec{b}$. \\
The direction of $\vec{a} \times \vec{b}$ is perpendicular to the plane containing the parallelogram.[/latex]

Q: [latex]
The volume of the parallelepiped (box) given by vectors $\vec{a}$, $\vec{b}$, and $\vec{c}$ is given by
[/latex]
A: [latex]
The triple product $\vec{a} \cdot (\vec{b} \times \vec{c})$, which is also given by the determinant
$$
\begin{vmatrix}
u_1 & u_2 & u_3 \\
v_1 & v_2 & v_3 \\
w_1 & w_2 & w_3
\end{vmatrix}
$$
as shown in the picture
[/latex]

Q: [latex]The parametric equation for a line through $\vec{p}$ in the direction of $\vec{v}$ is [/latex]
A: [latex]
$$L(t) = \vec{p} + t \vec{v}.$$
This picture shows a line in $\mathbb{R}^3$ \\
through $\vec{a} = (x_0,y_0,z_0)$ in
the \\ direction of $\vec{b}$:
[/latex]

Q: [latex]
The vector equation for a plane through $\vec{p}$ with normal vector $\vec{n} = (a,b,c)$ is
[/latex]
A: [latex]
$$\left< \vec{n}, \vec{x} \right>  = \left<\vec{n}, \vec{p}\right>$$
or
$$a x + b y + c z = d,$$
where $d = \left< \vec{n}, \vec{p} \right>$
[/latex]

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

Q: [latex]The relationship between length and dot product is[/latex]
A: [latex]
$$|\vec{v}| = \sqrt{\left<v,v\right>}$$
[/latex]

Q: [latex]
$$\left<\vec{a},\vec{b} + \vec{c}\right> = $$
[/latex]
A: [latex]
$$\left<\vec{a},\vec{b} + \vec{c}\right> = \left<\vec{a}, \vec{b}\right> + \left< \vec{a},\vec{c} \right>$$
[/latex]

Q: [latex]
$$\left< \lambda \vec{a}, \vec{b} \right> = $$
[/latex]
A: [latex]
$$\left< \lambda \vec{a}, \vec{b} \right> = \lambda \left< \vec{a}, \vec{b} \right> = \left< \vec{a}, \lambda \vec{b} \right> $$
[/latex]