我們來看看向量相乘的三種方法。首先，我們來看看向量的標量乘法。然後，我們看兩個向量相乘。我們將學習兩種不同的乘向量的方法，使用標量積和叉積。

如何用標量乘向量

當你將一個向量乘以一個標量時，向量的每個分量都會乘以這個標量。

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.

如何求兩個向量的標量積

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  . 

那麼，

.

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.  . 

如何求兩個向量的叉積

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

與標量積不同，向量積或叉積給出了一個向量作為答案。上面的公式給出了向量的大小。為了得到這個向量的方向，想象一下把螺絲刀從第一個向量的方向轉向第二個向量的方向。螺絲刀“進去”的方向是向量積的方向。

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  .

關於單位向量，我們有

另外，我們還有

對於分量，向量積由下式給出：，

例子

Find the cross product between vectors  and  .

.