Vector Algebra and Spatial Analytic Geometry

This chapter systematically studies basic vector operations, equations of points, lines, and planes in space, and their relationships, laying the foundation for spatial geometry and physical modeling.

Basic Concepts and Operations of Vectors

Definition of Vector

A vector is a quantity with both magnitude and direction, usually represented by a directed line segment.

Linear Operations of Vectors

Addition, subtraction, scalar multiplication
Linear combination

Coordinate Representation of Vectors

$\vec{a} = (a_1, a_2, a_3)$
Zero vector, unit vector

Dot Product (Scalar Product) of Vectors

$\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$
$|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$
$\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta$
Perpendicular: $\vec{a} \cdot \vec{b} = 0$

Cross Product (Vector Product) of Vectors

$\vec{a} \times \vec{b}$
The result is a vector perpendicular to both $\vec{a}$ and $\vec{b}$
$|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \sin\theta$
Parallel: $\vec{a} \times \vec{b} = 0$

Mixed Product of Vectors

$[\vec{a}, \vec{b}, \vec{c}] = (\vec{a} \times \vec{b}) \cdot \vec{c}$
Represents the volume of the parallelepiped formed by the three vectors

Points, Lines, and Planes in Space

Coordinates of a Point

$P(x_0, y_0, z_0)$

Equations of a Line

Symmetric form: $\frac{x - x_0}{l} = \frac{y - y_0}{m} = \frac{z - z_0}{n}$
Parametric form: $x = x_0 + lt,\ y = y_0 + mt,\ z = z_0 + nt$

Equations of a Plane

General form: $Ax + By + Cz + D = 0$
Point-normal form: $(x - x_0)A + (y - y_0)B + (z - z_0)C = 0$
Normal vector: $(A, B, C)$

Relationships between Points, Lines, and Planes

Conditions for parallelism, perpendicularity, intersection
Angle between two lines, angle between a line and a plane, angle between two planes

Distance Formulas

Distance from a point to a plane: $d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}}$
Distance from a point to a line

Introduction to Quadric Surfaces

Sphere, ellipsoid, paraboloid, hyperboloid, etc.
General equation: $Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0$

Exercises

Given $\vec{a} = (1,2,3)$ , $\vec{b} = (4,0,-1)$ , find $\vec{a} \cdot \vec{b}$ and $\vec{a} \times \vec{b}$ .
Find the parametric equations of the line passing through $A(1,2,3)$ with direction vector $(1,-1,2)$ .
Find the normal vector of the plane $x + 2y - 2z + 3 = 0$ and write its point-normal form equation.
Find the distance from the point $P(2,1,-1)$ to the plane $x - y + 2z - 4 = 0$ .
Determine whether the vectors $\vec{a} = (1,2,3)$ and $\vec{b} = (2,4,6)$ are parallel.
State the geometric meaning of the sphere $x^2 + y^2 + z^2 = 9$ .

Reference Answers

1. Given $\vec{a} = (1,2,3)$ , $\vec{b} = (4,0,-1)$ , find $\vec{a} \cdot \vec{b}$ and $\vec{a} \times \vec{b}$

$\vec{a} \cdot \vec{b} = 1\times4 + 2\times0 + 3\times(-1) = 4 - 3 = 1$

$\vec{a} \times \vec{b} = (2\times(-1) - 3\times0, 3\times4 - 1\times(-1), 1\times0 - 2\times4) = (-2, 13, -8)$

2. Find the parametric equations of the line passing through $A(1,2,3)$ with direction vector $(1,-1,2)$

$x = 1 + t$

y = 2 - t

$z = 3 + 2t$

3. Find the normal vector of the plane $x + 2y - 2z + 3 = 0$ and write its point-normal form equation

The normal vector is $(1,2,-2)$ .

Point-normal form: $(x - x_0) + 2(y - y_0) - 2(z - z_0) = 0$ , where $(x_0, y_0, z_0)$ is a point on the plane.

4. Find the distance from $P(2,1,-1)$ to the plane $x - y + 2z - 4 = 0$

$d = \frac{|2 - 1 + 2\times(-1) - 4|}{\sqrt{1^2 + 1^2 + 2^2}} = \frac{|2 - 1 - 2 - 4|}{\sqrt{1 + 1 + 4}} = \frac{|-5|}{\sqrt{6}} = \frac{5}{\sqrt{6}}$

5. Determine whether $\vec{a} = (1,2,3)$ and $\vec{b} = (2,4,6)$ are parallel

$\vec{b} = 2\vec{a}$ , so they are parallel.

6. State the geometric meaning of the sphere $x^2 + y^2 + z^2 = 9$

A sphere centered at the origin with radius $3$ .