Generally, we choose to establish a coordinate system when solving three-dimensional geometry problems, because it is safer and has a fixed routine. Many times these questions require you to calculate the normal vector (normal vector) of a certain surface . This is also a fixed method in high school. What we want to introduce here is a more advanced and faster method, which is to introduce the vector fork Multiplication (cross product, "vector" is the same as the concept of "vector" in physics, I have never figured out why mathematics and physics use different names, both in English are vector).
We have all learned the scalar product of vectors, which is the so-called dot product (dot product). After the scalar product of two vectors, a scalar is obtained. Here we define a new vector operation, which is vector product or cross product:
The result of the operation is still a vector, we record it as vector c , and its modulus is defined as:
Where θ is the angle between vector a and vector b , as shown in the figure below, the modulus of c is the area of a parallelogram with a and b as two sides.
The direction of c is defined as being perpendicular to the plane formed by a and b , and a , b and c form the right-hand spiral rule , that is, the direction of the four fingers of the right hand turns from a to b , and the thumb gets the direction of c . After defining this new operation, we can prove the following important properties from this definition:
The first two properties can be seen at a glance according to the definition, and the third distributive law of cross multiplication is a very important property, and the proof is also difficult, so we do not intend to go into details.
Then in the three-dimensional coordinate system, after a and b have coordinate representations, how to calculate the coordinates of c ? Note that the unit vectors in the positive directions of the x, y, and z axes are i , j , and k respectively , then:
Then according to the above three properties of the cross product, we get c as:
In the second equal sign we use property 3, in the third equal sign we use property 1 and property 2, and in the last equal sign we use the simple cross product relationship between i , j , k . This calculation formula allows us to directly obtain the coordinate representation of c from the coordinate representation of a and b , and the determinant can be expressed more concisely and is easy to remember:
With this formula, for the normal vector of any surface, we can always select two non-collinear vectors on the surface to directly cross-multiply them. Generally, in solving problems, we directly select the unit vectors on the two sides of the surface That's it.