📚 Solving Problems with Vectors in i and j Notation | 用 i 与 j 表示法解向量题
In Edexcel A-Level Mathematics, vector problems are often expressed using the standard unit vectors i and j. This notation makes it quick to add, subtract, scale and compare vectors, and it is especially useful for solving geometric and mechanical problems in two dimensions.
在 Edexcel A-Level 数学中，向量问题常用标准单位向量 i 和 j 表示。这种表示法能快速完成向量的加减、数乘与比较，对于求解二维几何与力学问题尤其有用。
1. Understanding i and j Base Vectors | 认识 i 与 j 基向量
The vectors i and j are unit vectors pointing in the positive x-direction and positive y-direction respectively. Any two-dimensional vector can be written uniquely as a combination ai + bj, where a and b are scalars.
向量 i 和 j 分别是沿 x 轴正方向和 y 轴正方向的单位向量。任何二维向量都可以唯一地写成 ai + bj 的组合，其中 a 与 b 是标量。
This means i = 1i + 0j and j = 0i + 1j, so they form a basis for the plane. A vector such as 5i – 2j is read as “5 units in the i direction and negative 2 units in the j direction”.
这意味着 i = 1i + 0j，j = 0i + 1j，因此它们构成了平面的一个基。像 5i – 2j 这样的向量可以读作“沿 i 方向 5 个单位，沿 j 方向负 2 个单位”。
2. Writing Vectors in Component Form | 将向量写成坐标形式
A vector such as v = 3i – 4j can also be written as the column vector (3, -4) in component form. The coefficient of i is the horizontal component, and the coefficient of j is the vertical component.
例如向量 v = 3i – 4j 也可写成坐标形式 (3, -4)。i 的系数是水平分量，j 的系数是垂直分量。
Converting between column vectors and i-j notation is usually the first step in exam problems, because each form suits different operations. The table below shows three vectors in both forms.
在考试题中，第一步通常是在列向量与 i-j 表示法之间进行转换，因为不同形式适合不同运算。下表展示了三个向量在两种形式下的表示。
| i-j notation | Column vector |
|---|---|
| 3i + 4j | (3, 4) |
| -2i + 5j | (-2, 5) |
| 0i – 7j | (0, -7) |
Always keep the signs attached to each component when converting, because losing a negative sign is a common source of error.
转换时一定要保留每个分量前面的符号，因为丢失负号是常见的错误来源。
3. Magnitude of a Vector | 向量的模长
The magnitude or length of a vector v = ai + bj is found using Pythagoras’ theorem:
向量 v = ai + bj 的模长（长度）可用勾股定理求得：
|v| = √(a² + b²)
This gives the distance from the initial point to the terminal point of the vector. It is always a non-negative scalar.
它表示从向量起点到终点的距离，并且总是一个非负标量。
For example, the vector 3i – 4j has magnitude √(3² + (-4)²) = √25 = 5. If a vector is given as 6i + 8j, its magnitude is √(6² + 8²) = √100 = 10.
例如，向量 3i – 4j 的模长为 √(3² + (-4)²) = √25 = 5。若向量为 6i + 8j，其模长为 √(6² + 8²) = √100 = 10。
Sometimes you are given the magnitude and one component and asked to find the other. For example, if |v| = 13 and v = 5i + kj, then √(5² + k²) = 13, so 25 + k² = 169, giving k² = 144 and k = ±12.
有时题目会给出模长和一个分量，要求另一个分量。例如，若 |v| = 13 且 v = 5i + kj，则 √(5² + k²) = 13，于是 25 + k² = 169，得到 k² = 144，所以 k = ±12。
4. Direction and Angle with the i Direction | 方向以及与 i 方向的夹角
The direction of a vector can be measured as the angle θ it makes with the positive i direction. Using trigonometry:
向量的方向可以用它与 i 正方向之间的夹角 θ 来表示。利用三角函数：
tan θ = b / a
for a vector ai + bj, provided a ≠ 0. This angle is usually measured anticlockwise from the positive x-axis.
对于向量 ai + bj，有上述关系，其中 a ≠ 0。这个角度通常从 x 轴正方向逆时针测量。
Always check the quadrant when finding direction, because the same tangent value can occur in two different quadrants. For example, vectors 3i + 4j and -3i – 4j have the same tangent ratio 4/3, but their directions differ by 180°.
求方向时一定要检查象限，因为同一个正切值可能出现在两个不同的象限中。例如，向量 3i + 4j 与 -3i – 4j 的正切比都是 4/3，但它们的方向相差 180°。
5. Adding, Subtracting and Scaling Vectors | 向量的加法、减法与数乘
To add or subtract vectors in i-j notation, combine the i coefficients and the j
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