8.1 Vector Concepts, Addition, Subtraction, and Scalar Multiplication

8.1 Vector Concepts, Addition, Subtraction, and Scalar Multiplication

Plane vectors serve as a fundamental bridge between geometry and algebra in the Gaokao Mathematics curriculum. By combining physical intuition about magnitude and direction with rigorous algebraic axioms, vector analysis provides powerful tools for solving geometric, trigonometric, and analytical problems.

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1. Fundamental Concepts of Plane Vectors

1.1 Definition and Representation

A vector in the plane is a quantity characterized by both a magnitude (length or norm) and a direction. Graphically, a vector is represented by a directed line segment $\vec{AB}$, where point $A$ is the initial point (tail) and point $B$ is the terminal point (head).

• Magnitude: The length of the directed line segment $\vec{AB}$, denoted by $|\vec{AB}|$ or $|\vec{a}|$. Magnitudes are non-negative real numbers ($|\vec{a}| \ge 0$).
• Free Vectors: In Euclidean plane geometry, vectors are considered free vectors. Two directed line segments with different spatial positions represent the exact same vector as long as they have identical length and identical direction.

1.2 Special Types of Vectors

Critical Gaokao Nuance: The zero vector $\vec{0}$ is parallel to every vector $\vec{a}$. Thus, $\vec{0} \parallel \vec{a}$ is universally true for all plane vectors. However, when using the scalar multiplier $\vec{b} = \lambda \vec{a}$, we must explicitly assume $\vec{a} \neq \vec{0}$ to guarantee uniqueness of $\lambda$.

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2. Vector Addition and Subtraction

2.1 Vector Addition

Vector addition is defined geometrically using two equivalent constructions:

1. Triangle Rule: To compute $\vec{a} + \vec{b}$, place the initial point of $\vec{b}$ at the terminal point of $\vec{a}$. The sum $\vec{a} + \vec{b}$ is the directed line segment from the initial point of $\vec{a}$ to the terminal point of $\vec{b}$. $$\vec{AB} + \vec{BC} = \vec{AC}$$
2. Parallelogram Rule: Place the initial points of $\vec{a}$ and $\vec{b}$ at a common origin $O$, so $\vec{OA} = \vec{a}$ and $\vec{OB} = \vec{b}$. Complete the parallelogram $OACB$. The diagonal vector $\vec{OC}$ starting from $O$ represents $\vec{a} + \vec{b}$.

Algebraic Properties of Addition

For any plane vectors $\vec{a}, \vec{b}, \vec{c}$:

• Commutative Law: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$
• Associative Law: $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$
• Identity Law: $\vec{a} + \vec{0} = \vec{a}$
• Inverse Law: $\vec{a} + (-\vec{a}) = \vec{0}$

Triangle Inequality for Vectors

For any plane vectors $\vec{a}$ and $\vec{b}$:

$$||\vec{a}| - |\vec{b}|| \le |\vec{a} + \vec{b}| \le |\vec{a}| + |\vec{b}|$$

• Equality $|\vec{a} + \vec{b}| = |\vec{a}| + |\vec{b}|$ holds if and only if $\vec{a}$ and $\vec{b}$ are codirectional (same direction) or at least one is $\vec{0}$.
• Equality $|\vec{a} + \vec{b}| = ||\vec{a}| - |\vec{b}||$ holds if and only if $\vec{a}$ and $\vec{b}$ are antidirectional (opposite direction) or at least one is $\vec{0}$.

2.2 Vector Subtraction

Vector subtraction is defined as the addition of the opposite vector:

$$\vec{a} - \vec{b} = \vec{a} + (-\vec{b})$$

Head-Minus-Tail Rule

When vectors share a common initial point $O$, such that $\vec{OA} = \vec{a}$ and $\vec{OB} = \vec{b}$, the difference $\vec{a} - \vec{b}$ is represented by the directed segment from the terminal point of $\vec{b}$ to the terminal point of $\vec{a}$:

$$\vec{OA} - \vec{OB} = \vec{BA} \quad \text{or} \quad \vec{OB} - \vec{OA} = \vec{AB}$$

Memory device: Subtracting vector points to the minuend (terminal minus initial).

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3. Scalar Multiplication and Collinearity Criteria

3.1 Scalar Multiplication

For a real scalar $\lambda \in \mathbb{R}$ and vector $\vec{a}$, the product $\lambda \vec{a}$ is a vector satisfying:

1. Magnitude: |$\lambda \vec{a}| = |\lambda| \cdot |\vec{a}|$.
2. Direction:
   • If $\lambda > 0$, $\lambda \vec{a}$ has the same direction as $\vec{a}$.
   • If $\lambda < 0$, $\lambda \vec{a}$ has the opposite direction to $\vec{a}$.
   • If $\lambda = 0$ or $\vec{a} = \vec{0}$, $\lambda \vec{a} = \vec{0}$.

Algebraic Laws of Scalar Multiplication

• Associative Law: $\lambda (\mu \vec{a}) = (\lambda \mu) \vec{a}$
• Distributive Laws: $(\lambda + \mu) \vec{a} = \lambda \vec{a} + \mu \vec{a}$ and $\lambda (\vec{a} + \vec{b}) = \lambda \vec{a} + \lambda \vec{b}$

3.2 Collinear Vector Theorem (Parallelism Condition)

Theorem (Vector Collinearity): Vector $\vec{b}$ is parallel to a non-zero vector $\vec{a}$ ($\vec{b} \parallel \vec{a}$) if and only if there exists a unique real scalar $\lambda \in \mathbb{R}$ such that:

$$\vec{b} = \lambda \vec{a}$$

3.3 Three-Point Collinearity Theorem

Theorem (Three-Point Collinearity): Points $A, B, C$ are collinear if and only if for any given origin point $O$, the position vector $\vec{OC}$ can be written as a linear combination of $\vec{OA}$ and $\vec{OB}$:

$$\vec{OC} = x \vec{OA} + y \vec{OB} \quad \text{with constraint} \quad x + y = 1$$

Alternatively, placing $C$ on line segment $AB$ with division ratio $\vec{AC} = k \vec{AB}$ gives $\vec{OC} = (1-k)\vec{OA} + k\vec{OB}$.

Proof of Three-Point Collinearity:

Points $A, B, C$ are collinear $\iff \vec{AC} \parallel \vec{AB} \iff \exists k \in \mathbb{R}$ such that $\vec{AC} = k \vec{AB}$. Expressing in terms of origin $O$:

$$\vec{OC} - \vec{OA} = k (\vec{OB} - \vec{OA}) \implies \vec{OC} = (1 - k) \vec{OA} + k \vec{OB}$$

Setting $x = 1 - k$ and $y = k$, we get $\vec{OC} = x \vec{OA} + y \vec{OB}$ where $x + y = (1 - k) + k = 1$. $\blacksquare$

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4. Fundamental Theorem of Plane Vectors (Basis Theorem)

Fundamental Theorem: If $\vec{e}_1$ and $\vec{e}_2$ are two non-collinear vectors in the same plane, then for any vector $\vec{a}$ in this plane, there exists a unique pair of real numbers $\lambda_1, \lambda_2 \in \mathbb{R}$ such that:

$$\vec{a} = \lambda_1 \vec{e}_1 + \lambda_2 \vec{e}_2$$

• The ordered pair of non-collinear vectors $\{\vec{e}_1, \vec{e}_2\}$ is called a basis for the plane.
• $\lambda_1 \vec{e}_1 + \lambda_2 \vec{e}_2$ is called the linear combination of $\vec{a}$ with respect to the basis $\{\vec{e}_1, \vec{e}_2\}$.

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5. Worked Gaokao Exam Examples

Worked Example 1: Intersection Ratio using Vector Linear Combinations

Problem (Gaokao Standard): In $\triangle ABC$, point $D$ is the midpoint of side $BC$. Point $E$ lies on side $AC$ such that $AE = 2 EC$. The line segments $AD$ and $BE$ intersect at point $P$. Express $\vec{AP}$ in terms of basis vectors $\vec{AB}$ and $\vec{AC}$, and determine the ratio $AP : PD$.

Solution:

1. Choose $\{\vec{AB}, \vec{AC}\}$ as the basis for the plane of $\triangle ABC$.
2. Since $D$ is the midpoint of $BC$, we have: $$\vec{AD} = \frac{1}{2} \vec{AB} + \frac{1}{2} \vec{AC}$$
3. Point $P$ lies on segment $AD$, so there exists scalar $\lambda \in (0, 1)$ such that $\vec{AP} = \lambda \vec{AD}$: $$\vec{AP} = \frac{\lambda}{2} \vec{AB} + \frac{\lambda}{2} \vec{AC} \quad \text{--- (Equation 1)}$$
4. On the other hand, points $B, P, E$ are collinear. Since $AE = 2 EC$, we have $\vec{AE} = \frac{2}{3} \vec{AC}$. Using the three-point collinearity theorem for line $BE$, there exists $\mu \in (0, 1)$ such that: $$\vec{AP} = (1 - \mu) \vec{AB} + \mu \vec{AE} = (1 - \mu) \vec{AB} + \frac{2}{3} \mu \vec{AC} \quad \text{--- (Equation 2)}$$
5. By the uniqueness clause of the Fundamental Basis Theorem, coefficients of $\vec{AB}$ and $\vec{AC}$ in Equation 1 and Equation 2 must be equal: $$\begin{cases} \frac{\lambda}{2} = 1 - \mu \\[6pt] \frac{\lambda}{2} = \frac{2}{3} \mu \end{cases}$$
6. Equating the two expressions for $\frac{\lambda}{2}$: $$1 - \mu = \frac{2}{3} \mu \implies \frac{5}{3} \mu = 1 \implies \mu = \frac{3}{5}$$
7. Substituting $\mu = \frac{3}{5}$ back gives $\frac{\lambda}{2} = 1 - \frac{3}{5} = \frac{2}{5} \implies \lambda = \frac{4}{5}$.
8. Therefore: $$\vec{AP} = \frac{2}{5} \vec{AB} + \frac{2}{5} \vec{AC}$$ Since $\vec{AP} = \frac{4}{5} \vec{AD}$, point $P$ divides $AD$ in ratio $AP : PD = 4 : 1$.

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