向量xiàngliàng

Core Concept

A vector is a quantity that has both magnitude and direction. Geometrically, vectors are represented by directed line segments; algebraically, they are represented by coordinates.

Vector Notation

1. Geometric Notation

Directed line segment $\overrightarrow{AB}$ where $A$ is the initial point and $B$ is the terminal point.

2. Letter Notation

Lowercase letters with arrows: $\vec{a}$, $\vec{b}$, $\vec{c}$

3. Coordinate Notation

2D vector: $\vec{a} = (x, y)$ or $\vec{a} = (a_1, a_2)$

3D vector: $\vec{a} = (x, y, z)$ or $\vec{a} = (a_1, a_2, a_3)$

Basic Concepts

Magnitude (Length)

For vector $\vec{a} = (x, y)$, magnitude is denoted $|\vec{a}|$ or $\|\vec{a}\|$:

$|\vec{a}| = \sqrt{x^2 + y^2}$

Zero Vector

Vector with magnitude 0, denoted $\vec{0} = (0, 0)$, with arbitrary direction

Unit Vector

Vector with magnitude 1, denoted $\vec{e}$

Unit vector in direction of $\vec{a}$: $\vec{e} = \frac{\vec{a}}{|\vec{a}|}$

Equal Vectors

Vectors with same direction and magnitude

Opposite Vectors

Vectors with opposite directions but equal magnitudes. Opposite of $\vec{a}$ is $-\vec{a}$

Vector Operations

1. Vector Addition

Geometric Meaning

Parallelogram rule or triangle rule

Coordinate Operation

$\vec{a} + \vec{b} = (x_1, y_1) + (x_2, y_2) = (x_1 + x_2, y_1 + y_2)$

Properties

• Commutative: $\vec{a} + \vec{b} = \vec{b} + \vec{a}$
• Associative: $(\vec{a} + \vec{b}) + \vec{c} = \vec{a} + (\vec{b} + \vec{c})$

2. Vector Subtraction

$\vec{a} - \vec{b} = (x_1, y_1) - (x_2, y_2) = (x_1 - x_2, y_1 - y_2)$

Geometric meaning: Points from terminal of $\vec{b}$ to terminal of $\vec{a}$

3. Scalar Multiplication

$k\vec{a} = k(x, y) = (kx, ky)$

where $k$ is a real number.

Properties:

• $k > 0$: Same direction as $\vec{a}$, magnitude $k|\vec{a}|$
• $k < 0$: Opposite direction to $\vec{a}$, magnitude $|k||\vec{a}|$
• $k = 0$: Results in zero vector

4. Dot Product (Scalar Product)

$\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$

where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.

Coordinate Form

$\vec{a} \cdot \vec{b} = (x_1, y_1) \cdot (x_2, y_2) = x_1x_2 + y_1y_2$

Important Properties

• $\vec{a} \cdot \vec{b} = 0 \Leftrightarrow \vec{a} \perp \vec{b}$ (vectors perpendicular)
• $\vec{a} \cdot \vec{a} = |\vec{a}|^2$

Real-World Applications

Application 1: Displacement

Problem: Person walks 3km east then 4km north from origin. Find displacement magnitude and direction.

Solution: Displacement vector: $\vec{s} = (3, 4)$

Magnitude: $|\vec{s}| = \sqrt{3^2 + 4^2} = 5 \text{ km}$

Direction: Angle with east axis $\tan\theta = \frac{4}{3}, \quad \theta \approx 53.1°$

Answer: 5km at 53.1° north of east

Application 2: Force Composition

Problem: Forces $\vec{F_1} = (3, 0)$ N and $\vec{F_2} = (0, 4)$ N act at same point. Find resultant.

Solution: $\vec{F} = \vec{F_1} + \vec{F_2} = (3, 4) \text{ N}$

Magnitude: $|\vec{F}| = \sqrt{3^2 + 4^2} = 5 \text{ N}$

Application 3: Velocity Decomposition

Problem: Plane flies at 200 km/h toward northeast (45° from east). Find east and north components.

Solution: $v_x = 200\cos45° = 100\sqrt{2} \approx 141.4 \text{ km/h}$ $v_y = 200\sin45° = 100\sqrt{2} \approx 141.4 \text{ km/h}$

CSCA Practice Problems

💡 Note: The following practice problems are designed based on the CSCA exam syllabus and Chinese standardized test formats to help students familiarize themselves with question types and problem-solving approaches.

Example 1: Basic (Difficulty ★★☆☆☆)

Given vector $\vec{a} = (3, 4)$, find $|\vec{a}|$.

Options:

• A. 3
• B. 4
• C. 5
• D. 7

Solution: $|\vec{a}| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$

Answer: C

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Example 2: Intermediate (Difficulty ★★★☆☆)

Given $\vec{a} = (2, 1)$ and $\vec{b} = (1, -3)$, find $\vec{a} + 2\vec{b}$.

Solution:

$2\vec{b} = 2(1, -3) = (2, -6)$ $\vec{a} + 2\vec{b} = (2, 1) + (2, -6) = (4, -5)$

Answer: $(4, -5)$

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Example 3: Advanced (Difficulty ★★★★☆)

Given $\vec{a} = (1, 2)$ and $\vec{b} = (2, -1)$, find:

1. $\vec{a} \cdot \vec{b}$
2. Angle $\theta$ between $\vec{a}$ and $\vec{b}$

Solution:

(1) Dot product: $\vec{a} \cdot \vec{b} = 1(2) + 2(-1) = 0$

(2) Angle: Since $\vec{a} \cdot \vec{b} = 0$: $\vec{a} \perp \vec{b}$ $\theta = 90°$

Answers: (1) 0 (2) 90° (perpendicular)

Common Mistakes

❌ Mistake 1: Equal vectors must have same initial point

Correction: Vectors are equal if they have same direction and magnitude, regardless of position. Vectors can be translated.

❌ Mistake 2: Magnitude can be negative

Correction: Magnitude (length) is always non-negative: $|\vec{a}| \geq 0$.

❌ Mistake 3: Dot product result is a vector

Correction: Dot product result is a scalar (number), not a vector.

❌ Mistake 4: Zero vector has no direction

Correction: Zero vector direction is arbitrary, can be parallel to any vector.

❌ Mistake 5: Wrong angle range

Correction: Angle $\theta$ between two non-zero vectors ranges from $[0°, 180°]$ or $[0, \pi]$.

Study Tips

1. ✅ Understand essence: Both magnitude and direction, different from scalars
2. ✅ Master coordinate operations: Addition, subtraction, scalar multiplication, dot product
3. ✅ Grasp geometric meaning: Visualize vector operations in coordinate system
4. ✅ Remember perpendicularity: $\vec{a} \cdot \vec{b} = 0 \Leftrightarrow \vec{a} \perp \vec{b}$
5. ✅ Apply to real world: Displacement, force, velocity are all vectors

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💡 Exam Tip: Vectors are essential in CSCA geometry and physics. Coordinate operations and dot product are frequently tested!