Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版：英文原文完整保留，中文紧随对应单元；术语采用 v20260916 确认表，自动语义审校结果可追溯。

Basic Vector Operations

基本矢量运算
Both a magnitude and a direction must be specified for a vector quantity, in contrast to a scalar quantity which can be quantified with just a number. Any number of vector quantities of the same type (i.e., same units) can be combined by basic vector operations.
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矢量量必须同时指定大小和方向，这与仅需一个数值即可量化标量量形成对比。相同类型的矢量量（即相同单位）可以通过基本的矢量运算进行组合。注意！这是一份大型HTML文档。你需要等待它完全加载，以便所有上面的链接都能正常运作。
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Math of vectors
向量的索引
 
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Graphical Vector Addition

矢量图解法

Adding two vectors A and B graphically can be visualized like two successive walks, with the vector sum being the vector distance from the beginning to the end point. Representing the vectors by arrows drawn to scale, the beginning of vector B is placed at the end of vector A. The vector sum R can be drawn as the vector from the beginning to the end point.

将两个矢量A和B相加可以形象地看作是两次连续的行走，矢量和就是从起点到终点的矢量距离。用箭头按比例绘制矢量，矢量B的起点放在矢量A的终点处。矢量和R可以画成从起点到终点的矢量。

The process can be done mathematically by finding the components of A and B, combining to form the components of R, and then converting to polar form.

这个过程可以通过找到A和B的分量，将它们组合成R的分量，然后转换为极坐标形式来数学地完成。
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Vector concepts
索引矢量概念
 
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Example of Vector Components

矢量的示例

Finding the components of vectors for vector addition involves forming a right triangle from each vector and using the standard triangle trigonometry.

为矢量相加求分量，需将每个矢量构成一个直角三角形，并利用标准三角形三角函数进行计算。

The vector sum can be found by combining these components and converting to polar form.

矢量和可以通过将这些分量组合并转换为极坐标形式来找到。
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Vector concepts
索引矢量概念
 
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Polar Form Example

极坐标形式示例

After finding the components for the vectors A and B, and combining them to find the components of the resultant vector R, the result can be put in polar form by

在找到矢量A和B的分量，并将它们结合起来求出 resultant vector R 的分量之后，结果可以表示为极坐标形式

Some caution should be exercised in evaluating the angle with a calculator because of ambiguities in the arctangent on calculators.

在使用计算器评估角度时应谨慎，因为计算器上的反正切函数存在歧义。
Index

Vector concepts
索引矢量概念
 
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Combining Vector Components

矢量分量

After finding the components for the vectors A and B, these components may be just simply added to find the components of the resultant vector R.

在找到矢量A和B的分量后，这些分量可以简单相加以求得 resultant vector R 的分量。

The components fully specify the resultant of the vector addition, but it is often desirable to put the resultant in polar form.

矢量相加的合成矢量完全由其分量指定，但通常希望将结果表示为极坐标形式。
Index

Vector concepts
索引矢量概念
 
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Resolving a Vector Into Components

将矢量分解为分量

Vectors are resolved into components by use of the triangle trig relationships. You may change the length or angle of the polar form of the vector, and the components will be calculated below.

矢量通过三角三角函数关系分解为分量。您可以改变矢量的极坐标形式的长度或角度，分量将在下方计算。
For vector A=
at angle degrees,

the horizontal component is
=
and the vertical component is
=
The input to the boxes for units is arbitrary; they serve to emphasize that the process of vector addition is independent of the units of the vector.
对于矢量 A 与角度 degrees 成角，其水平分量是 =，垂直分量是 =。输入到方框中的单位是任意的；它们的作用是强调矢量加法的过程与矢量的单位无关。
中文译文中的待填/计算数值依次对应：1：r 2：u1 3：ag 4：x 5：u2 6：y 7：u3。实际数值以上方原输入框为准。
Note: this Javascript routine does not work for angle exactly equal to 90° .

注意：这个JavaScript程序在角度恰好为90°时无法正常工作。
 Index

Vector concepts
索引矢量概念
 
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Magnitude and Direction from Components

由分量确定的大小和方向

If the components of a vector are known, then its magnitude and direction can be calculated with the use of the Pythagorean relationship and triangle trig. This is called the polar form of the vector.

若已知矢量的分量，则可通过勾股定理和三角函数计算其大小和方向。这称为矢量的极坐标形式。
If the horizontal component is
=
and the vertical component is
= ,

then the magnitude is
=
and the angle is
= degrees.

The input to the boxes for units is arbitrary; they serve to emphasize that the process of vector addition is independent of the units of the vector.

框内的单位输入是任意的；它们的作用是强调矢量相加的过程与矢量的单位无关。

Some caution should be exercised in evaluating the angle with a calculator because of ambiguities in the arctangent on calculators.

在使用计算器评估角度时应谨慎，因为计算器上的反正切函数存在歧义。
若水平分量为 = ，垂直分量为 = ，则大小为 = ，角度为 = 度。
中文译文中的待填/计算数值依次对应：1：x 2：u1 3：y 4：u2 5：r 6：u3 7：ag。实际数值以上方原输入框为准。
Index

Vector concepts
索引矢量概念
 
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Vector Addition, Two Vectors

矢量相加，两个矢量

Vector addition involves finding vector components, adding them and finding the polar form of the resultant.

矢量加法涉及找到矢量的分量，相加并求出结果的极坐标形式。
The addition of vector
A= at degrees,
and vector
B= at degrees,

yields components:

+ =

+ =
The resultant has magnitude
R =
and angle
= degrees.
矢量 A 在 30 度方向，矢量 B 在 60 度方向，其分量为：A_x = B_x = 1.0，A_y = B_y = 1.0。结果矢量的大小 R = 2.0，角度 = 45 度。
中文译文中的待填/计算数值依次对应：1：a 2：aa 3：b 4：ab 5：ax 6：bx 7：rx 8：ay 9：by 10：ry 11：r 12：angr。实际数值以上方原输入框为准。
Note: There are some combinations of angles and values, like those where an exactly zero result is anticipated or an angle of exactly 90 degrees, where the Javascript routine is unstable and gives erroneous values. Usually you can try values very close to such conditions to test the accuracy.
注意：有些角度和数值的组合，例如预期结果恰好为零或角度恰好为90度的情况，JavaScript程序会不稳定并给出错误的数值。通常你可以尝试接近这些条件的值来测试准确性。
 Index

Vector concepts
索引矢量概念
 
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Vector Addition, Three Vectors

矢量加法，三个矢量

Vector addition involves finding vector components, adding them and finding the polar form of the resultant.

矢量加法涉及找到矢量的分量，相加并求出结果的极坐标形式。
The addition of vectors
A= at degrees,
B= at degrees,and
C= at degrees

yields components:

+ + =

+ + =
The resultant has magnitude
R =
and angle
= degrees.
向量A、B和C分别以度、度和度加入后，其分量为：+ + = + + = 。结果的大小R = ，角度=度。
中文译文中的待填/计算数值依次对应：1：a 2：aa 3：b 4：ab 5：c 6：ac 7：ax 8：bx 9：cx 10：rx 11：ay 12：by 13：cy 14：ry 15：r 16：angr。实际数值以上方原输入框为准。
Index

Vector concepts
索引矢量概念
 
HyperPhysics***** Mechanics
力学
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Vector Addition, Four Vectors

矢量加法，四矢量
Vector addition involves finding vector components, adding them and finding the polar form of the resultant.
The addition of vectors
A= at degrees,
B= at degrees,
C= at degrees,
D= at degrees,

yields components:

+++=

+++=
The resultant has magnitude
R =
and angle
= degrees.
矢量相加涉及找到矢量的分量，相加并求出结果的极坐标形式。将矢量A= at degrees, B= at degrees, C= at degrees, D= at degrees相加，得到分量：+ + + = + + + = 结果的大小R = 和角度= degrees。
中文译文中的待填/计算数值依次对应：1：a 2：aa 3：b 4：ab 5：c 6：ac 7：d 8：ad 9：ax 10：bx 11：cx 12：dx 13：rx 14：ay 15：by 16：cy 17：dy 18：ry 19：r 20：angr。实际数值以上方原输入框为准。
Index

Vector concepts
索引矢量概念
 
HyperPhysics***** Mechanics
力学
R Nave
 Go Back
返回