\( \textnormal{The distance between points } P_1 (x_1, y_1, z_1) \textnormal{ and } P_2 (x_2, y_2, z_2) \textnormal{ is } \) \begin{equation} |\, P_1 \, P_2 \, | = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \qquad (\textnormal{Distance formula in 3 dimensions}) \tag{1.1} \end{equation} \( \textnormal{An equation of a sphere with center } C(j, p, w) \textnormal { and radius } r \textnormal{ is } \) \begin{equation} (x - j)^2 + (y - p)^2 + (z - w)^2 = r^2 \qquad (\textnormal{Equation of a sphere}) \tag{1.2} \end{equation}
If a and b are vectors where the point of a is at the end of b, then the sum a + b is the vector from the initial point of a to the end of b (Definition of vector addition) (1.3)
If c is a scalar and a is a vector, then the multiple c * a is the vector whose length is | c | times the length of a. The direction of c * a is the same as a if c > 0 and opposite a if c < 0. If c = 0, c * a = 0. (Definition of scalar multiplication) (1.4)
\( \textnormal{Given the points } P(x_1, y_1, z_1) \textnormal { and } Q(x_2, y_2, z_2), \textnormal{ the vector } \textbf{p} \textnormal{ with representation } \vec{PQ} \textnormal{ is } \) \begin{equation} \textbf{p} = \langle x_2 - x_1, y_2 - y_1, z_2 - z_1 \rangle \tag{1.5} \end{equation}
\( \textnormal{If } \textbf{s} = \langle s_1, s_2, s_3 \rangle \textnormal { and } \textbf{k} = \langle k_1, k_2, k_3 \rangle, \textnormal{ then the dot product of } \textbf{s} \textnormal{ and } \textbf{k} \textnormal{ is } \) \begin{equation} \textbf{s} \cdot \textbf{k} = s_1k_1 + s_2k_2 + s_3k_3 \tag{1.6} \end{equation}
\( \textnormal{If } \textbf{s} = \langle s_1, s_2, s_3 \rangle \textnormal { and } \textbf{k} = \langle k_1, k_2, k_3 \rangle, \textnormal{ then the cross product of } \textbf{s} \textnormal{ and } \textbf{k} \textnormal{ is the vector } \) \begin{equation} \textbf{s} \times \textbf{k} = \langle s_2k_3 - s_3k_2, s_3k_1 - s_1k_3, s_1k_2 - s_2k_1 \rangle \tag{1.7} \end{equation}
\begin{equation} \textbf{r} = \textbf{r}_0 + t \textbf{v} \qquad (\textnormal{Vector equation of a line}) \tag{1.8} \end{equation}
Say that f and v are differentiable vector functions, g is a real-valued function, and k is a scalar. Then
\begin{equation} \frac{d}{dt} [\textbf{f}(t) + v(t)] = \textbf{f} \,'(t) + \textbf{v} \,'(t) \tag{1} \end{equation}