📚 Solving Problems with Vectors Using i and j Notation | 利用 i 和 j 表示法求解向量问题
Vectors written in terms of the standard unit vectors i and j appear throughout Edexcel A‑Level Mathematics – from pure geometry to mechanics. Being fluent with i and j notation allows you to add, subtract, and scale vectors effortlessly, calculate magnitudes, check for parallel directions, and solve real‑world problems involving displacement, velocity, and acceleration. This article brings together the essential techniques and typical exam‑style problems you need to master.
在 Edexcel A‑Level 数学中，用标准单位向量 i 和 j 表示的向量贯穿纯数与力学。熟练掌握 i 和 j 符号，你就能轻松地进行向量加减、数乘、求模、判断平行方向，并解决位移、速度、加速度等实际问题。本文汇总了必备技巧和典型考题，助你彻底攻克这类问题。
1. Vector Basics with i and j | 用 i 和 j 表示的向量基础
In two dimensions, any vector can be expressed uniquely as a = xi + yj, where x and y are real numbers. The vector i points one unit in the positive x-direction, and j points one unit in the positive y-direction. So the vector a = 3i − 2j represents a displacement of 3 units right and 2 units down.
在二维空间中，任何向量均可唯一地写成 a = xi + yj 的形式，其中 x 和 y 为实数。向量 i 指向正 x 轴方向一个单位，j 指向正 y 轴方向一个单位。因此向量 a = 3i − 2j 表示向右 3 单位、向下 2 单位的位移。
The components x and y are called the i-component and j-component. They are exactly the same as the entries in a column vector: a = ( x y ) as a column. Writing vectors in i,j form often makes algebra clearer and reduces mistakes when working with multiple vectors.
分量 x 和 y 分别称为 i 分量和 j 分量，它们与列向量的对应项完全一致。用 i、j 形式表示向量，能使代数运算更加直观，在处理多个向量时可有效减少错误。
2. Addition, Subtraction, and Scalar Multiplication | 向量的加法、减法与标量乘法
To add two vectors, simply add their i-components and their j-components separately. If a = a₁i + a₂j and b = b₁i + b₂j, then a + b = (a₁ + b₁)i + (a₂ + b₂)j. Subtraction works in the same way: a − b = (a₁ − b₁)i + (a₂ − b₂)j.
两个向量相加，只需分别相加它们的 i 分量和 j 分量。若 a = a₁i + a₂j，b = b₁i + b₂j，则 a + b = (a₁ + b₁)i + (a₂ + b₂)j。减法同理：a − b = (a₁ − b₁)i + (a₂ − b₂)j。
When multiplying a vector by a scalar k, each component is multiplied by k: ka = k(a₁i + a₂j) = (k a₁)i + (k a₂)j. This changes the length of the vector without altering its direction, unless k is negative, in which case the direction is reversed.
向量乘以标量 k 时，每个分量都乘以 k：ka = k(a₁i + a₂j) = (k a₁)i + (k a₂)j。这会改变向量的长度而不改变其方向，除非 k 为负，此时方向反转。
3. Magnitude and Direction | 向量的模与方向
The magnitude (or length) of a = xi + yj is found using Pythagoras' theorem:
向量 a = xi + yj 的模（长度）由勾股定理求得：
|a| = √x² + y²
For example, p = 3i − 4j has magnitude √(3² + (−4)²) = √9 + 16 = 5.
例如，p = 3i − 4j 的模为 √(3² + (−4)²) = √9 + 16 = 5。
The direction of a vector is often given as an angle measured from the positive i-direction (east). If a vector makes an angle θ with the positive x-axis, then tan θ = y/x. Always draw a sketch to decide the correct quadrant, because the arctan function on your calculator only gives an angle between −90° and 90°.
向量的方向通常以从正 i 方向（东）量起的角度表示。若向量与正 x 轴的夹角为 θ，则 tan θ = y/x。务必画出草图判断正确的象限，因为计算器中的反正切函数只给出 −90° 到 90° 之间的角度。
4. Unit Vectors | 单位向量
A unit vector has magnitude 1 and points in the same direction as a given vector. To find the unit vector in the direction of a, divide the vector by its magnitude:
单位向量的模为 1，方向与给定向量相同。求与 a 同方向的单位向量，只需将向量除以其模：
â = a / |a| = (x/| a |)i + (y/| a |)j
For instance, the unit vector alongside v = 6i − 8j is (6/10)i − (8/10)j = 0.6i − 0.8j. Checking: (0.6)² + (−0.8)² = 1.
例如，与 v = 6i − 8j 同方向的单位向量为 (6/10)i − (8/10)j = 0.6i − 0.8j。检验：(0.6)² + (−0.8)² = 1。
Unit vectors are particularly useful when you need a direction but not a specific length, such as when finding a force of a given magnitude acting along a certain line.
当需要方向而非特定长度时，单位向量尤其有用，例如求沿某一直线作用、大小为给定值的力。
5. Position Vectors and Displacement | 位置向量与位移向量
If a point P has coordinates (x, y), its position vector relative to the origin O is OP = xi + yj. The vector that takes you from point A to point B is the displacement vector AB, given by OB − OA.
若点 P 的坐标为 (x, y)，则它相对于原点 O 的位置向量为 OP = xi + yj。从点 A 到点 B 的向量称为位移向量 AB，可由 OB − OA 求得。
Always remember: AB = b − a where a and b are the position vectors of A and B. This simple subtraction is the key to countless geometric problems.
务必记住：AB = b − a，其中 a 和 b 分别是 A 和 B 的位置向量。这个简单的减法是无数的几何问题的关键。
6. Parallel Vectors and Collinearity | 平行向量与共线性
Two vectors u and v are parallel if one is a scalar multiple of the other: v = ku for some real k. In components, v₁i + v₂j = k(u₁i + u₂j). Therefore the ratios v₁/u₁ and v₂/u₂ are equal (provided no denominator is zero).
若两个向量互为标量倍数，即 v = ku，则它们平行。用分量表示时，有 v₁i + v₂j = k(u₁i + u₂j)，因此比值 v₁/u₁ 与 v₂/u₂ 相等（分母非零时）。
Three points A, B, C are collinear (lie on the same straight line) if the vectors AB and BC are parallel, which also means AB = λ BC for some scalar λ. Exam questions often ask you to find an unknown coordinate by imposing collinearity.
若 AB 与 BC 平行，即 AB = λ BC，则 A、B、C 三点共线。考题常要求通过共线条件求出未知坐标。
7. Using Vectors to Solve Geometric Problems | 利用向量解决几何问题
Vectors shine in plane geometry. For a quadrilateral ABCD, you can prove it is a parallelogram by showing that AB = DC. To show that a point M is the midpoint of AB, use the average of position vectors: m = (a + b)/2.
向量在平面几何中大放异彩。对于四边形 ABCD，可通过证明 AB = DC 来证明它是平行四边形。若要证明点 M 是 AB 的中点，可利用位置向量的平均值：m = (a + b)/2。
Partition ratios also arise. If a point P divides AB in the ratio m:n, then p = (na + mb)/(m + n). Work carefully with the order – always draw a diagram and label the section ratio from the first point to the dividing point, then to the second point.
分割比问题也很常见。若点 P 以 m:n 的比分割 AB，则 p = (na + mb)/(m + n)。处理顺序时务必谨慎——始终画出示意图，标注从第一个点到分点、再到第二个点的分割比。
Vector proofs of medians, centroids, and parallel lines are common. For instance, in triangle OAB, if C is the midpoint of OA and D the midpoint of OB, then CD = ½ AB, proving line CD is parallel to AB.
中线、重心与平行线的向量证明十分常见。例如，在三角形 OAB 中，若 C 为 OA 中点，D 为 OB 中点，则 CD = ½ AB，从而证明 CD 平行于 AB。
8. Kinematics with Constant Acceleration Vectors | 恒定加速度下的运动学向量
In mechanics, the position, velocity, and acceleration of a particle moving in a plane are written using i and j. For constant acceleration a, the standard SUVAT equations apply to each component separately:
在力学中，质点在平面上运动的位置、速度和加速度均用 i 和 j 表示。对于恒定加速度 a，标准的 SUVAT 方程可分别应用于每个分量：
v = u + at, r = r₀ + ut + ½ at²
Here u is initial velocity, v final velocity, a acceleration, r position at time t, and r₀ initial position. By equating i- and j-components, you can solve for time, speed, or distance. For example, if v = (3t − 2)i + (4 − t)j, the particle is moving parallel to i when the j-component is zero.
其中 u 为初速度，v 为末速度，a 为加速度，r 为 t 时刻的位置，r₀ 为初始位置。通过对应 i 分量和 j 分量，可求解时间、速率或距离。例如，若 v = (3t − 2)i + (4 − t)j，当 j 分量为零时，质点平行于 i 运动。
9. Problem-Solving Strategies and Common Pitfalls | 解题策略与常见错误
When tackling vector problems: always convert given information into i,j form; draw a clear diagram; write down position vectors and displacement vectors explicitly; check for parallel or perpendicular conditions if needed; and solve component equations separately.
解决向量问题时：始终将已知信息转化为 i、j 形式；画出清晰的示意图；明确写出位置向量和位移向量；必要时检查平行或垂直条件；并分别求解分量方程。
Common mistakes include: forgetting that AB = b − a (not a − b); mixing up magnitude and direction; using degrees instead of radians when finding a direction bearing; and misreading division ratios. Also, never assume that two vectors are perpendicular just because their dot product is zero – at A‑Level you must use the scalar (dot) product, where u·v = u₁ v₁ + u₂ v₂, and perpendicular means u·v = 0.
常见错误包括：忘记 AB = b − a（而非 a − b）；混淆模与方向；在求解方向角时错误使用度数而非弧度；误读分割比。另外，切勿仅凭点积为零就判定两向量垂直——在 A‑Level 中你需要使用标量积：<
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