向量的定義

向量是一個量值（大小）和方向。幾何學上，向量可以用有向線段來表示，線段的方向指向向量的方向，線段的長度與向量的大小成正比。

如何寫向量

A vector can be written down in several ways. One method is to use bold characters e.g.  . You can also use an underline ( ) or an arrow drawn on top of a letter ( ). If the symbol for a vector is written without these, it is taken to be the magnitude of the vector.

Two vectors that have the same length and direction are equal. In the diagram below,  .

如何找到向量的分量

若要在給定方向上找到向量的分量，請畫一條與所需方向平行的線，穿過向量的“尾”端。然後，從向量的“鼻尖”上畫一條垂直線到這條線上。向量在給定方向上的分量是，從向量的“尾”到垂直線的直線長度。

For example, on the diagram below, the component of the vector  along the  -axis is  and the component along the  -axis is  . 

從三角學，我們有：

and,

Generally, if a vector with magnitude makes an angle  to a given direction, then the component of the vector along that direction is  , and the component of the vector in the direction perpendicular to that direction is  .

例子

一架飛機以每小時253公里的速度起飛，與跑道成15度角。假設太陽直射頭頂，求出飛機陰影沿跑道的速度。

The speed of the shadow is the component of the plane’s velocity along the runway. Since the plane is travelling at an angle of 15o to the runway, the speed of the shadow is then  km h-1.

相反，如果一個向量沿兩個垂直方向的分量已知，我們可以用簡單的三角學來求向量沿其中一個方向的夾角，我們還可以計算原始向量的大小。

例子

A lawnmower is pushed along the ground, with a force  exerted along the handle. The vertical and horizontal components of the force are 30.6 N and 25.7 N respectively. Find a) size of the force    and b) the angle  that the lawnmower makes with the ground.

首先，為了找到力的大小，我們使用畢達哥拉斯定理：

N.

The angle is given by 

如何在笛卡爾坐標系中表示向量

If the components of a vector  along the  ,  and  axes are  ,  and  respectively, the vector can be written as  .

如何求向量的大小

Magnitude refers to the size of the vector, without taking into c***ideration its direction. The magnitude of a vector  is written as  . If the letter is simply written as  , this is also taken to indicate the magnitude of the vector.

If a vector  , then its magnitude .

例子

The electric field vector at a point is given by  N C-1. Find the magnitude of the electric field.

N C-1.

什麼是單位向量(unit vectors)？

A unit vector is a vector with a magnitude of 1 unit. Unit vectors are often written with a ‘hat’ above the letter. e.g.  . The unit vector along the direction of a vector  , is defined as:

In particular, on the Cartesian coordinate system, the unit vectors along the  ,  and   axes are written as , and respectively.

Using these unit vectors, a vector in the 3-dimensional Cartesian coordinate system can be written as a sum of 3 vectors along the  ,  and  directi***. This is done by taking components of the vector along  ,  and  axes, and multiplying each component by the unit vector of the corresponding axis.

For example, the vector  can be written as  .

如何加減向量