If we have to find that the definition of the quantity product of plane vectors has a geometric meaning, then we can say that the quantity product of a vector is equal to the weighted extension of one of the vector "modules".
It is the length of one vector projected on another vector as a weighted value, multiplied by the length of the projected vector itself.
It is a product, which is intuitively represented as a number.
Because it uses the concept of projection in its definition, it has natural use cases when dealing with problems containing angles between two straight lines, such as the proof of the cosine theorem of triangles, the proof of the equation of the difference identity transformation in trigonometry, and the determination of the plane angle size of a dihedral angle in a three-dimensional figure.
Let's expand on it a little bit:
For example, for the cosine theorem, let a, b, and c be the three sides of a triangle, and a, b, and c are the corresponding three angles.
If the three sides of the triangle are represented by vectors, there is the following relation:
Both sides are squared at the same time
Expand on both sides:
The cosine theorem is proven.
This proof method is much more intuitive than the geometric proof method.
That said, the definition of the product of vector quantities gives us a more concise tool for solving problems with angles of straight lines.
Another example is the proof of the cosine formula of the sum difference in trigonometric identity transformations.
The cosine identity looks like this:
If the proof is to be proved by geometric methods, the formula between the two points must be used
Simplification can finally lead to the following:
Since there is actually an angular relationship between OA and OB, we try to use the quantity product formula of the vector to prove that:
We now think of OA and OB as two vectors:
Both sides are square:
The above vectors are represented by coordinates and descended into the collation:
The result is the same as using the two-point formula, but it is significantly easier to use than the geometric method, because you no longer have to rotate the alpha minus beta to the initial position.
Similarly, in three-dimensional graphics, when calculating the cosine or sine value of the plane angle in the dihedral angle, if the geometric method is used, it is cumbersome, not to mention, the main reason is that it is too unintuitive, and it will really make people crazy if you want to find the plane angle of the dihedral angle. In this case, the simple way is to find the normal vectors of the two faces, and then calculate the trigonometric value of the angle by calculating the quantitative product of the two normal vectors.
How do I calculate the normal vector of a plane?That's the problem that the outer product of the vector is trying to solve.
Of course, there are other methods available in high school, and you don't have to figure out the problem of vector products in middle school.
Others, such as judging the perpendicular problems between two straight lines, parallel problems, etc., are all small cases, and they are just routine operations, so I will not talk about them here.
Thank you for reading.