我们来看看向量相乘的三种方法。首先，我们来看看向量的标量乘法。然后，我们看两个向量相乘。我们将学习两种不同的乘向量的方法，使用标量积和叉积。

如何用标量乘向量

当你将一个向量乘以一个标量时，向量的每个分量都会乘以这个标量。

Suppose we have a vector  , that is to be multiplied by the scalar  . Then, the product between the vector and the scalar is written as  . If  , then the multiplication would increase the length of  by a factor  .  If  , then, in addition to increasing the magnitude of  by a factor  , the direction of the vector would also be reversed.

With regards to vector components, each component gets multiplied by the scalar. For instance, if a vector  , then  .

例子

The momentum vector  of an object is given by  , where  is the mass of the object and  is the velocity vector. For an object with a mass of 2 kg having a velocity of  m s-1, find the momentum vector.

The momentum is kg m s-1.

如何求两个向量的标量积

The scalar product (also known as the dot product) between two vectors and is written as . This is defined as,

where  is the angle between the two vectors if they are placed tail-to-tail as shown below:

The scalar product between two vectors yields a scalar quantity. Geometrically, this quantity is equal to the product of the magnitude of one vector’s projection on the other and the magnitude of the “other” vector:

Using the components of vectors along the Cartesian plane, we could obtain the scalar product as follows. If the vector  and  , then the scalar product

例子

Vector  and  . Find  .

例子

The work done by a force , when it causes a displacement  for an object is given by, . Suppose a force of   N causes a body to move, whose displacement under the force is is  m. Find the work done by the force.

J.

例子

Find the angle between the two vectors  and .

From the definition of the scalar product, .  Here, we have  and  . 

那么，

.

If two vectors are perpendicular to each other, then the angle  between them is 90o. In this case,  and so the scalar product becomes 0. In particular, for unit vectors in the Cartesian coordinate system, we note that,

For parallel vectors, the angle  between them is 0o. In this case,  and the scalar product simply becomes the products of the magnitudes of the vectors. In particular,

The scalar product is commutative. i.e.  .

The scalar product is also distributive. i.e.  . 

如何求两个向量的叉积

The cross product (also known as the vector product) between two vectors and is written as . This is defined as,

与标量积不同，向量积或叉积给出了一个向量作为答案。上面的公式给出了向量的大小。为了得到这个向量的方向，想象一下把螺丝刀从第一个向量的方向转向第二个向量的方向。螺丝刀“进去”的方向是向量积的方向。

For instance, in the above diagram, the vector product is  will point into the page, whereas  will point out of the page.

Clearly, then, vector product is not commutative. Rather,  .

The vector product between two parallel vectors is 0. This is because the angle  between them is 00, making the  .

关于单位向量，我们有

另外，我们还有

对于分量，向量积由下式给出：，

例子

Find the cross product between vectors  and  .

.