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向量夹角为锐角的充要条件
英文回答：
To determine the necessary and sufficient condition for
the angle between two vectors to be acute, we need to
consider the dot product of the two vectors. The dot
product of two vectors is defined as the product of their
magnitudes and the cosine of the angle between them.
Let's consider two vectors, A and B, with magnitudes
|A| and |B|, respectively. The dot product of A and B is
given by A · B = |A| |B| cosθ, where θ is the angle
between the two vectors.
Now, for the angle between the vectors to be acute, we
want cosθ to be positive. This means that the dot product
A · B must be positive. Therefore, the necessary and
sufficient condition for the angle between two vectors to
be acute is that their dot product is positive.
In other words, if the dot product of two vectors A and
B is positive, then the angle between them is acute.
Conversely, if the dot product is negative or zero, then
the angle between them is obtuse or a right angle,
respectively.
Let's consider an example to illustrate this. Suppose
we have two vectors A = (2, 3) and B = (4, 1). To find the
dot product, we multiply the corresponding components and
sum them up: A · B = (2 4) + (3 1) = 8 + 3 = 11.
Since the dot product A · B is positive (11 > 0), we
can conclude that the angle between the vectors A and B is
acute.
中文回答：
要确定两个向量夹角为锐角的必要和充分条件，我们需要考虑
这两个向量的点积。两个向量的点积被定义为它们的模的乘积和它
们之间夹角的余弦值的乘积。
让我们考虑两个向量A和B，它们的模分别为|A|和|B|。向量A
和B的点积由A · B = |A| |B| cosθ给出，其中θ是两个向量
之间的夹角。
现在，为了使两个向量之间的夹角为锐角，我们希望cosθ为
正数。这意味着向量的点积A · B必须为正数。因此，两个向量之
间夹角为锐角的必要和充分条件是它们的点积为正数。
换句话说，如果两个向量A和B的点积为正数，则它们之间的
夹角为锐角。相反，如果点积为负数或零，则它们之间的夹角为钝
角或直角。
让我们举一个例子来说明。假设我们有两个向量A = (2, 3)和
B = (4, 1)。要计算点积，我们将对应的分量相乘并求和，A · B
= (2 4) + (3 1) = 8 + 3 = 11。
由于点积A · B为正数（11 > 0），我们可以得出结论，向量
A和B之间的夹角为锐角。
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