Lecture Note: §12 Vectors and the Geometry of Space

Published: April 09, 2026

Last Update: 2026-04-09

Three-Dimensional Coordinate Systems
- The direction of the $z$-axis is determined by the right-hand rule
Distance in Three Dimensions $\lvert P_1 P_2\rvert = \sqrt{(x_2 - x_1)^{2} + (y_2 - y_1)^{2} + (z_2 - z_1)^{2}}$
- Equation of a Sphere: $(x - x_0)^{2} + (y - y_0)^{2} + (z - z_0)^{2} = a^{2}$

Definition: The vector represented by the directed line segment $\overrightarrow{AB}$ has initial point $A$ and terminal point $B$, and its length is denoted by $\lvert\overrightarrow{AB}\rvert$. Two vectors are equal if they have the same length and direction.
- The component form of $\vec{v}$ is $\vec{v} = \langle v_1, v_2 \rangle\quad\text{or}\quad \vec{v} = \langle v_1, v_2, v_3 \rangle.$
- If $\vec{v} \neq \vec{0}$, then
  - $\dfrac{\vec{v}}{\lvert\vec{v}\rvert}$ is a unit vector called the direction of $\vec{v}$;
  - the equation $\vec{v} = \lvert\vec{v}\rvert \dfrac{\vec{v}}{\lvert\vec{v}\rvert}$ expresses $\vec{v}$ as its length times its direction.
The magnitude or length of the vector $\vec{v}$ $\lvert \vec{v}\rvert = \sqrt{v_1^2 + v_2^2 + v_3^2}$
Vector Operations
- Addition: $\vec{u} + \vec{v} = \langle u_1 + v_1,\ u_2 + v_2,\ u_3 + v_3 \rangle$
- Scalar multiplication: $k\vec{u} = \langle k u_1,\ k u_2,\ k u_3 \rangle$
- Properties of Vector Operations
Unit Vectors. A vector $\vec{v}$ of length $1$ is called a unit vector. The standard unit vectors are $\vec{i} = \langle 1, 0, 0 \rangle, \quad \vec{j} = \langle 0, 1, 0 \rangle, \quad \text{and} \quad \vec{k} = \langle 0, 0, 1 \rangle.$
The midpoint $M$ of the line segment joining points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ is the point $\left( \frac{x_1 + x_2}{2},\ \frac{y_1 + y_2}{2},\ \frac{z_1 + z_2}{2} \right).$

Dot Product $\vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 + u_3 v_3.$
- The angle $\theta$ between two nonzero vectors $\mathbf{u}$ and $\mathbf{v}$ is $\theta = \arccos\left( \frac{\mathbf{u} \cdot \mathbf{v}}{|\mathbf{u}| |\mathbf{v}|} \right).$
- $\vec{u} \cdot \vec{v} = \lvert \vec{u}\rvert \lvert \vec{v}\rvert \cos\theta$.
- Properties of the Dot Product
Orthogonal. Vectors $\vec{u}$ and $\vec{v}$ are orthogonal if $\vec{u} \cdot \vec{v} = 0$.
The vector projection of $\vec{u}$ onto $\vec{v}$ is the vector $\operatorname{proj}_{\vec{v}} \vec{u} = \left( \dfrac{\vec{u} \cdot \vec{v}}{\lvert\vec{v}\rvert^{2}} \right) \vec{v} = \left( \dfrac{\vec{u} \cdot \vec{v}}{\lvert\vec{v}\rvert} \right) \dfrac{\vec{v}}{\lvert\vec{v}\rvert}.$

Cross Product. $\vec{u} \times \vec{v} = (\lvert\vec{u}\rvert \lvert\vec{v}\rvert \sin \theta)\ \vec{n}.$
- $\lvert \vec{u} \times \vec{v}\rvert = \lvert\vec{u}\rvert \lvert\vec{v}\rvert \sin \theta$ is the area of a parallelogram
- Nonzero vectors $\vec{u}$ and $\vec{v}$ are parallel if and only if $\vec{u} \times \vec{v} = \vec{0}$.
- Properties of the Cross Product
  1. $(r\vec{u}) \times (s\vec{v}) = (rs)(\vec{u} \times \vec{v})$
  2. $\vec{u} \times (\vec{v} + \vec{w}) = \vec{u} \times \vec{v} + \vec{u} \times \vec{w}$
  3. $\vec{v} \times \vec{u} = -(\vec{u} \times \vec{v})$
  4. $(\vec{v} + \vec{w}) \times \vec{u} = \vec{v} \times \vec{u} + \vec{w} \times \vec{u}$
  5. $\vec{0} \times \vec{u} = \vec{0}$
  6. $\vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{u} \cdot \vec{v})\vec{w}$
- Calculating the Cross Product as a Determinant $\vec{u} \times \vec{v} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix}.$
Triple Scalar or Box Product: $(\vec{u} \times \vec{v}) \cdot \vec{w}$
- The absolute value of this product is the volume of the parallelepiped.
- Calculating the Triple Scalar Product as a Determinant$(\vec{u} \times \vec{v}) \cdot \vec{w} = \begin{vmatrix} u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \end{vmatrix}$ —

Equation for a Line
- Vector Equation: $\vec{r}(t) = \vec{r}_0 + t\vec{v}, \quad -\infty < t < \infty,$
- Parametric Equation: $x = x_0 + tv_1, \quad y = y_0 + tv_2, \quad z = z_0 + tv_3, \quad -\infty < t < \infty.$
- Symmetric Equation: $\frac{x - x_0}{v_1} = \frac{y-y_0}{v_2} = \frac{z - z_0}{v_3}$
- Line of Intersection of Two Plane$\begin{cases}A_1x + B_1y + C_1z = D_1\\ A_2 x + B_2y + C_2z = D_2\end{cases}$
Distance from a Point $S$ to a Line Through $P$ Parallel to $\vec{v}$ $d = \frac{\lvert\overrightarrow{PS} \times \vec{v}\rvert}{\lvert\vec{v}\rvert}$
Equation for a Plane in Space.
- Vector equation: $\vec{n} \cdot \overrightarrow{P_0P} = 0$
- Component equation: $A(x - x_0) + B(y - y_0) + C(z - z_0) = 0$
- Component equation simplified: $Ax + By + Cz = D, \quad \text{where} \quad D = Ax_0 + By_0 + Cz_0$
Distance from a Point $S$ to a Plane Through a Point $P$ with a Normal $\vec{n}$ $d = \left\lvert \overrightarrow{PS} \cdot \frac{\vec{n}}{\lvert\vec{n}\rvert} \right\rvert$
- Distance between two parallel planes
- Distance between skew lines

Cylinder
- A cylinder is a surface that is generated by moving a line that is parallel to the given line along the curve, while keeping it parallel to the given line. The curve is called a generating curve for the cylinder.
Quadratic Surfaces
- A quadric surface is the graph in space of a second-degree equation in $x$, $y$, and $z$. The simplest equation $Ax^{2} + By^{2} + Cz^{2} + Dz = E,$ where $A$, $B$, $C$, $D$, and $E$ are constants. The basic quadric surfaces are ellipsoids, paraboloids, elliptical cones, and hyperboloids. Spheres are special cases of ellipsoids.