Direction Ratios and Coordinate Geometry | 方向比与坐标几何

📚 Direction Ratios and Coordinate Geometry | 方向比与坐标几何

Direction ratios are a fundamental concept in coordinate geometry, especially in three-dimensional space. They provide a simple way to describe the orientation of a line, and they connect naturally with vectors, dot products, and equations of lines and planes.

方向比是坐标几何中的一个基本概念，尤其在三维空间中应用广泛。它提供了一种简洁的方法来描述直线的方向，并与向量、点积以及直线和平面的方程自然联系起来。

1. What Are Direction Ratios? | 什么是方向比？

Direction ratios of a line are any three numbers \(a, b, c\) that are proportional to the direction cosines of the line. In other words, if the direction cosines are \(\cos \alpha, \cos \beta, \cos \gamma\), then there exists a positive constant \(k\) such that \(\cos \alpha : \cos \beta : \cos \gamma = a : b : c\).

一条直线的方向比是任意三个与这条直线的方向余弦成比例的数 \(a, b, c\)。也就是说，如果方向余弦为 \(\cos \alpha, \cos \beta, \cos \gamma\)，则存在正常数 \(k\)，使得 \(\cos \alpha : \cos \beta : \cos \gamma = a : b : c\)。

For a line with direction vector \(\mathbf{d} = (a, b, c)\), the numbers \(a, b, c\) are themselves direction ratios. They describe how much the line moves along each coordinate axis for a given step.

对于方向向量为 \(\mathbf{d} = (a, b, c)\) 的直线，三个坐标分量 \(a, b, c\) 本身就是方向比。它们描述了沿各坐标轴移动一步时，直线在各方向上的变化量。

2. Direction Cosines and Their Relation | 方向余弦及其关系

Direction cosines are the cosines of the angles that a line makes with the positive \(x\)-, \(y\)-, and \(z\)-axes. They are usually denoted by \(\ell, m, n\). If the line makes angles \(\alpha, \beta, \gamma\) with the axes, then \(\ell = \cos \alpha\), \(m = \cos \beta\), \(n = \cos \gamma\).

方向余弦是直线与 \(x\) 轴、\(y\) 轴、\(z\) 轴正方向所成角的余弦值，通常记为 \(\ell, m, n\)。若直线与坐标轴所成角分别为 \(\alpha, \beta, \gamma\)，则 \(\ell = \cos \alpha\)，\(m = \cos \beta\)，\(n = \cos \gamma\)。

A key identity holds for any line:

任何直线都满足以下关键恒等式：

\(\ell^2 + m^2 + n^2 = 1\)

If the direction ratios are \(a, b, c\), then the direction cosines can be recovered by normalising:

如果已知方向比为 \(a, b, c\)，则可以通过归一化得到方向余弦：

\(\ell = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad n = \frac{c}{\sqrt{a^2 + b^2 + c^2}}\)

Note that the denominator is simply the magnitude of the direction vector.

注意分母就是方向向量的模长。

3. Direction Ratios of a Line Segment | 线段的方向比

Given two points \(A(x_1, y_1, z_1)\) and \(B(x_2, y_2, z_2)\), the direction ratios of the line segment \(AB\) are simply the differences between the coordinates:

已知两点 \(A(x_1, y_1, z_1)\) 和 \(B(x_2, y_2, z_2)\)，线段 \(AB\) 的方向比就是两点的坐标差：

\((x_2 – x_1, \; y_2 – y_1, \; z_2 – z_1)\)

These direction ratios are independent of which endpoint you start from, provided you keep the orientation consistent. Reversing the direction gives the negatives: \((-a, -b, -c)\).

只要保持方向一致，方向比与从哪个端点出发无关。将方向反转则得到相反数：\((-a, -b, -c)\)。

For example, if \(A = (1, 2, 3)\) and \(B = (4, 6, 10)\), then direction ratios are \((3, 4, 7)\).

例如，若 \(A = (1, 2, 3)\)，\(B = (4, 6, 10)\)，则方向比为 \((3, 4, 7)\)。

4. Equation of a Line in 3D Using Direction Ratios | 用方向比表示三维直线方程

In three-dimensional coordinate geometry, a line is uniquely determined by a point and a direction. If a line passes through \(P_0(x_0, y_0, z_0)\) and has direction ratios \((a, b, c)\), then a general point \(P(x, y, z)\) on the line satisfies the parametric equations:

在三维坐标几何中，一条直线由一个点和方向唯一确定。若直线经过 \(P_0(x_0, y_0, z_0)\)，方向比为 \((a, b, c)\)，则直线上任意点 \(P(x, y, z)\) 满足参数方程：

\(x = x_0 + a t, \quad y = y_0 + b t, \quad z = z_0 + c t\)

Eliminating the parameter \(t\) gives the symmetric (Cartesian) form:

消去参数 \(t\) 得到对称式（笛卡尔式）方程：

\(\frac{x – x_0}{a} = \frac{y – y_0}{b} = \frac{z – z_0}{c}\)

Here \(a, b, c\) cannot all be zero. If one of them is zero, the corresponding numerator is set to zero instead of a fraction.

这里 \(a, b, c\) 不能全为零。如果其中一个为零，则相应的分子等于零，而不写成比例式。

5. Angle Between Two Lines | 两直线夹角

The angle \(\theta\) between two lines with direction ratios \((a_1, b_1, c_1)\) and \((a_2, b_2, c_2)\) is found using the dot product of their direction vectors:

两条方向比分别为 \((a_1, b_1, c_1)\) 和 \((a_2, b_2, c_2)\) 的直线夹角 \(\theta\) 可通过其方向向量的点积求得：

\(\cos \theta = \frac{a_1 a_2 + b_1 b_2 + c_1 c_2}{\sqrt{a_1^2 + b_1^2 + c_1^2} \, \sqrt{a_2^2 + b_2^2 + c_2^2}}\)

The two lines are perpendicular if and only if the numerator is zero:

两条直线垂直当且仅当分子为零：

\(a_1 a_2 + b_1 b_2 + c_1 c_2 = 0\)

The two lines are parallel if and only if their direction ratios are proportional:

两条直线平行当且仅当它们的方向比成比例：

\(\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}\)

This parallels the slope relationship in two dimensions.

这与二维坐标几何中的斜率关系完全对应。

6. Projection and Direction Ratios | 投影与方向比

The projection of a vector \(\mathbf{u}\) onto the direction of a line with direction ratios \((a, b, c)\) is given by the scalar projection divided by the magnitude of the direction vector. If \(\mathbf{u} = (u_1, u_2, u_3)\), then the scalar projection onto the line is:

向量 \(\mathbf{u}\) 在方向比为 \((a, b, c)\) 的直线方向上的投影，等于用方向向量的模去除标量投影。若 \(\mathbf{u} = (u_1, u_2, u_3)\)，则在该直线方向上的标量投影为：

\(\frac{u_1 a + u_2 b + u_3 c}{\sqrt{a^2 + b^2 + c^2}}\)

The vector projection is this scalar multiplied by the unit direction vector.

向量投影则等于该标量乘以单位方向向量。

This is useful for decomposing forces, velocities, or displacements along a given line in mechanics and physics.

这在力学和物理学中常用于将力、速度或位移分解到给定直线方向。

7. Coordinate Geometry: Straight Lines in 2D | 坐标几何：二维直线

In two dimensions, direction ratios reduce to a pair of numbers \((a, b)\), and they are directly related to the slope \(m = b/a\). The equation of a line through \((x_0, y_0)\) with direction ratios \((a, b)\) is:

在二维坐标几何中，方向比退化为一对数 \((a, b)\)，并且与斜率 \(m = b/a\) 直接相关。过点 \((x_0, y_0)\)、方向比为 \((a, b)\) 的直线方程为：

\(\frac{x – x_0}{a} = \frac{y – y_0}{b}\)

This is equivalent to the point-slope form \(y – y_0 = m(x – x_0)\) when \(a \neq 0\).

当 \(a \neq 0\) 时，这与点斜式 \(y – y_0 = m(x – x_0)\) 等价。

The direction ratios of a line are also the components of a direction vector, while the normal vector has components \((b, -a)\). This observation links direction ratios to the general form \(Ax + By + C = 0\).

方向比也是方向向量的分量，而法向量分量为 \((b, -a)\)。这一关系将方向比与一般式 \(Ax + By + C = 0\) 联系起来。

8. Distance, Section Formula, and Direction Ratios | 距离、分点公式与方向比

In 3D, the distance between two points is the magnitude of the vector connecting them:

在三维空间中，两点间距离等于连接它们的向量模长：

\(d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2 + (z_2 – z_1)^2}\)

The section formula for a point dividing a line segment in the ratio \(m : n\) internally can be expressed elegantly using direction ratios. If \(P\) divides \(A\) and \(B\) internally in the ratio \(m : n\), then the coordinates of \(P\) are:

内分点公式可以用方向比巧妙地表达。若点 \(P\) 内分 \(A\)、\(B\) 所成线段，且 \(AP : PB = m : n\)，则 \(P\) 的坐标为：

\(P = \left( \frac{n x_1 + m x_2}{m + n}, \; \frac{n y_1 + m y_2}{m + n}, \; \frac{n z_1 + m z_2}{m + n} \right)\)

This is equivalent to writing \(P = A + \frac{m}{m+n} (B – A)\), where \(B – A\) has the same direction ratios as the line.

这等价于 \(P = A + \frac{m}{m+n} (B – A)\)，其中 \(B – A\) 与直线方向比相同。

9. Applications in Planes | 平面中的应用

A plane can be described by a point and a normal vector, but direction ratios also play a role in plane geometry. For example, the angle between a line and a plane uses the direction ratio of the line and the normal vector of the plane.

平面可以由一点和法向量描述，但方向比在平面几何中也有重要应用。例如，直线与平面的夹角可以使用直线的方向比和平面的法向量来求解。

If a plane has normal vector \(\mathbf{n} = (A, B, C)\) and a line has direction ratios \((a, b, c)\), then the angle \(\theta\) between the line and its orthogonal projection on the plane satisfies:

若平面的法向量为 \(\mathbf{n} = (A, B, C)\)，直线方向比为 \((a, b, c)\)，则直线与其在平面上正投影的夹角 \(\theta\) 满足：

\(\sin \theta = \frac{|A a + B b + C c|}{\sqrt{A^2 + B^2 + C^2} \, \sqrt{a^2 + b^2 + c^2}}\)

This is because the angle between the line and the plane is the complement of the angle between the line and the normal vector.

因为直线与平面的夹角是直线与法向量夹角的余角。

10. Common Pitfalls and Exam Tips | 常见错误与考试技巧

A frequent error is confusing direction ratios with direction cosines. Remember that direction ratios are only proportional to direction cosines, not equal unless the vector is already of unit length.

一个常见错误是混淆方向比与方向余弦。请记住：方向比只是与方向余弦成比例，除非向量已经单位化，否则二者并不相等。

Another mistake is forgetting to check for parallel or perpendicular conditions. Always verify whether \(a_1 b_2 – a_2 b_1 = 0\) in 2D, or the proportional condition in 3D, before applying angle formulas.

另一个错误是忘记检查平行或垂直条件。在应用夹角公式前，应先在二维情形验证 \(a_1 b_2 – a_2 b_1 = 0\)，在三维情形验证比例条件。

When solving problems, use the following strategy:

解题时建议采用以下策略：

Extract the direction vector directly from the coordinates or equation.
直接根据坐标或方程提取方向向量。
Normalise the direction vector if you need direction cosines.
如果需要方向余弦，则将方向向量单位化。
Use the dot product for angles, not cross product.
求夹角用点积，不要用叉积。
Check whether the question asks for the acute angle or the obtuse angle.
注意题目要求的是锐角还是钝角。

Practice with past paper questions involving 3D lines and planes to become familiar with the standard formats.

多做涉及三维直线和平面的历年真题，熟悉标准题型。

Direction ratios are a unifying idea in coordinate geometry. They simplify the description of lines, angles, projections, and planes. Mastering them provides a solid foundation for higher-dimensional geometry and vector calculus.

方向比是坐标几何中的一个统一性概念。它简化了直线、夹角、投影和平面的描述。熟练掌握方向比为更高维几何和向量微积分奠定坚实基础。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Direction Ratios and Coordinate Geometry | 方向比与坐标几何