📚 Displacement Vectors in Two Dimensions | 二维位移向量
A vector is a quantity that possesses both magnitude and direction, in contrast to a scalar which only has magnitude. In physics and mathematics, displacement is a fundamental vector that describes the change in position of an object. When we work in two dimensions, displacement vectors allow us to model motion on a plane, such as the movement of a drone or a ship navigating a map. This chapter explores the representation, components, and operations on two-dimensional displacement vectors, building a toolkit essential for IB Mathematics and beyond.
向量是既有大小又有方向的量，而标量只有大小。在物理和数学中，位移是描述物体位置变化的基本向量。在二维情形下，位移向量帮助我们在平面上建立运动模型，比如无人机的飞行或船舶的航线。本章将探讨二维位移向量的表示、分量及运算，为IB数学及后续学习打造一套核心工具。
1. Vectors and Scalars | 向量与标量
A scalar quantity is fully described by a single number (with appropriate units). Examples include distance, speed, mass, and temperature. Scalars obey ordinary arithmetic rules.
标量可以用一个数字（加上适当单位）完整描述。例如距离、速率、质量和温度。标量遵循普通的算术运算法则。
In contrast, a vector quantity requires both a magnitude and a direction for its complete specification. Displacement, velocity, acceleration, and force are all vectors. The length of a displacement vector tells us how far an object has moved, and the arrowhead indicates the direction.
而向量则需要大小和方向两者来完整描述。位移、速度、加速度和力都是向量。位移向量的长度表示物体移动了多远，箭头指示方向。
2. Representing Displacement in Two Dimensions | 二维位移的表示
In a coordinate plane, a displacement vector can be drawn as an arrow from the initial point A to the terminal point B. The vector is denoted by AB with an arrow in geometric sketches, or by a single bold letter such as v. The displacement tells us the straight-line difference between two positions.
在坐标平面上，位移向量可以画成从起点A指向终点B的箭头。几何图上记为带箭头的AB，也可用单个粗体字母例如v表示。位移给出两个位置之间的直线差异。
If a particle moves from point A(2,1) to point B(5,4), the displacement vector in component form is AB 3) is interpreted as 3 units in the positive x-direction and 3 units in the positive y-direction.
如果质点从点A(2,1)移动到B(5,4)，位移向量的分量形式为AB = (5 − 2, 4 − 1) = (3, 3)。记法(3,3)的含义是沿正x方向移动3个单位，沿正y方向移动3个单位。
3. Components of a Vector | 向量的分量
Any two-dimensional vector can be expressed in terms of its horizontal and vertical components. We often write a vector v as v = (v₁, v₂) or v = vₓi + vᵧj, where i and j are unit vectors in the x and y directions respectively.
任何二维向量都可以用其水平和垂直分量表示。我们通常将向量v记为v = (v₁, v₂) 或 v = vₓi + vᵧj，其中i和j分别是沿x和y方向的单位向量。
The component form makes calculation straightforward: to find the displacement from P(x₁, y₁) to Q(x₂, y₂), we simply subtract the coordinates: PQ Here, the x-component is the horizontal change and the y-component is the vertical change.
分量形式使计算变得简单：从P(x₁, y₁)到Q(x₂, y₂)的位移只需相减坐标：PQ = (x₂ − x₁, y₂ − y₁)。其中x分量是水平变化量，y分量是竖直变化量。
4. Magnitude of a Vector | 向量的大小（模）
The magnitude (or length) of a displacement vector v = (a, b) is found using the Pythagorean theorem. The magnitude, denoted |v|, is given by |v| = √a² + b². For example, a displacement of (3, 4) has magnitude √3² + 4² = √9 + 16 = √25 = 5.
位移向量v = (a, b)的大小（或称模）可通过勾股定理求得。大小记作|v|，公式为|v| = √a² + b²。例如，位移(3, 4)的大小为√3² + 4² = √9 + 16 = √25 = 5。
Magnitude is always a non-negative scalar. It represents the shortest distance between the initial and final positions, regardless of the actual path taken. This property is central to solving navigation and physics problems.
大小永远是一个非负标量。它代表起点到终点之间的最短距离，与实际走过的路径无关。这一性质是解决导航和物理问题的核心。
5. Direction of a Vector | 向量的方向
n θ = b/a, but care must be taken with the quadrant to obtain the correct reference angle.
二维向量的方向通常用它从正x轴逆时针量起的角度θ表示。对向量(a, b)，可利用tan θ = b/a求方向角，但需注意所在象限以得到正确的参考角。
In navigation contexts, direction is often given as a bearing—a clockwise angle from north. For instance, a displacement of (3, 3) has direction 45° from the positive x-axis, but as a bearing from north it would be 045° if we align axes appropriately.
在导航场景中，方向常以方位角给出——即从正北顺时针测量的角度。例如位移(3, 3)从正x轴起的夹角为45°，若将x轴对齐正东，则相应的方位角为045°。
6. Vector Addition | 向量加法
Two displacement vectors can be added using the triangle rule or the parallelogram rule. Graphically, to add u and v, place the tail of v at the head of u; the resultant vector u + v goes from the tail of u to the head of v.
两个位移向量可用三角形法则或平行四边形法则相加。图解法中，要做u + v，只需将v的尾端接到u的首端；合成向量u + v从u的尾端指向v的首端。
In component terms, addition is simply done by adding corresponding components: u + v = (u₁ + v₁, u₂ + v₂). This is commutative, meaning u + v = v + u, a property that reflects geometric symmetry.
在分量形式上，加法只需将对应分量相加：u + v = (u₁ + v₁, u₂ + v₂)。加法满足交换律，即u + v = v + u，这一性质反映了几何上的对称性。
7. Vector Subtraction | 向量减法
Subtracting a vector is equivalent to adding its negative. The negative of a vector v has the same magnitude but opposite direction, written as −v = (−v₁, −v₂). Thus, u − v = u + (−v).
向量减法等价于加上减向量的负向量。向量v的负向量大小相同但方向相反，记作−v = (−v₁, −v₂)。因此u − v = u + (−v)。
Geometrically, u − v can be visualized by placing the tails of u and v together and drawing the arrow from the head of v to the head of u. This vector represents the relative displacement from the tip of v to the tip of u.
几何上，u − v可以将u和v的尾端重合，然后从v的首端向u的首端画箭头。这个向量表示从v端点到u端点的相对位移。
8. Scalar Multiplication | 标量乘法
When a displacement vector is multiplied by a scalar k, its magnitude is scaled by |k| on remains the same if k > 0 or reverses if k < 0. If k = 0, the result is the zero vector 0 = (0, 0).
位移向量乘以标量k时，其大小变为原来的|k|倍；若k>0方向不变，若k<0方向反转。k=0时得到零向量0 = (0, 0)。
In component form, kv ches or shrinks the vector. For example, doubling a velocity vector (2 m/s east, 3 m/s north) gives (4, 6) m/s, a faster speed along the same line of motion.
分量形式下，kv = (k×v₁, k×v₂)。这种运算会拉伸或缩短向量。例如，将速度向量(向东2 m/s，向北3 m/s)乘以2得到(4, 6) m/s，即沿同一运动方向以更快速度运动。
9. Unit Vectors and Position Vectors | 单位向量与位置向量
A unit vector has magnitude 1 and is often used to indicate direction. The basic unit vectors in 2D are i = (1, 0) and j = (0, 1). Any displacement can be written as a linear combination: v = ai + bj.
单位向量大小为1，常用于指示方向。二维基本单位向量为i = (1, 0) 和 j = (0, 1)。任何位移都可写成线性组合：v = ai + bj。
A position vector locates a point relative to the origin. The position vector of point P(x, y) is r = xi + yj. Displacement from P to Q can be expressed as r_Q − r_P, which links position vectors and displacement elegantly.
位置向量表示点相对于原点的位置。点P(x, y)的位置向量为r = xi + yj。从P到Q的位移可表示为r_Q − r_P，优美地将位置向量和位移联系了起来。
10. Applications of Displacement Vectors | 位移向量的应用
Displacement vectors are invaluable in navigation. A plane flying with a wind effect can be modelled by adding its air-speed vector to the wind's velocity vector. The resultant ground displacement over a given time is the vector sum.
位移向量在导航中无价。受风影响的飞机飞行可将其空速向量与风速向量相加来建模。在一定时间内对地的总位移就是这些向量的和。
In mechanics, the concept of relative motion uses displacement subtraction: the displacement of object A relative to B is r_A − r_B. This helps determine collision courses or interception paths in two dimensions.
力学中，相对运动的概念使用位移减法：物体A相对B的位移为r_A − r_B。这有助于确定二维平面中的碰撞路线或拦截路径。
Even in computing and graphics, translating a sprite across a screen is a direct application of adding displacement vectors to position vectors. Thus mastery of two-dimensional displacement vectors serves as a foundation for further studies in mathematics, physics, and engineering.
即使在计算机和图形学中，让精灵在屏幕上移动也是位置向量加上位移向量的直接应用。因此，掌握二维位移向量为数学、物理和工程学进一步深造奠定基础。
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