1. Vector

A vector is a quantity that has both magnitude and direction. In physics, a vector is also called a vector. (Corresponding to it is called a scalar, only the amount of size)

Generally, a directed line segment is used to represent a vector, as shown in the following figure:

When expressing vector geometry, add arrows above the letters, such as: $\underset{AB}{\rightarrow}$

When expressed in vector algebra, it is generally lowercase bold letters, such as: a , b , c, etc., but the bold letters cannot be clearly reflected in handwriting, so arrows are often added to the letters, such as: $\underset{a}{\rightarrow}$

In order to be able to operate vectors like numbers, we must also put the vectors in a certain coordinate system in mathematics: if the tails of all vectors in the space are pulled down to the origin of the coordinates, the n-dimensional point space can be compared with the n-dimensional Vector spaces establish a one-to-one correspondence. The point (0,0,0...) in the n-dimensional point space is taken as the origin, then each point can have a vector corresponding to it, and this vector is the vector starting from the origin of the coordinates to this point.

After determining the coordinate system, the vector and the point are in one-to-one correspondence, and the point is represented by an ordered array of coordinates, so the vector can also be represented by an ordered array of coordinates. Operates on vectors through ordered arrays of vectors. For example: in the two-dimensional coordinate system, $\underset{AB}{\rightarrow}$ =(4,5), the ordered array (4,5) is a vector $\underset{AB}{\rightarrow}$

2. Definition of inner product of vector

The inner product of vectors is also called scalar product, scalar product, and dot product.

There are two definitions of the inner product, as follows:

$a\cdot b=\begin{Vmatrix} a \end{Vmatrix} \begin{Vmatrix} b \end{Vmatrix}\cos\theta$ (where it $\left \| a \right \|$ represents the length of the vector a , and some also use it $\left | a \right |$ to represent the length)

The product of the lengths of vectors a and b times the cosine of the angle between them.

$a\cdot b=a_{1}b_{1}+a_{2}b_{2}+...+a_{n}b_{n}$

The coordinate components of vectors a and b correspond to the sum of the products, respectively.

When I first came into contact with the definition of the inner product of a vector, I wonder why there are two definitions. From the definition, it can be seen that the results of both definitions are a value. Then the final result of the different calculation methods given by these two definitions The values should be consistent, how to verify?

3. Geometric interpretation of vector inner product

Take the three-dimensional space as an example, take any two vectors a and b , coincide the tails of the two vectors with the origin, take the plane formed by the intersection of vectors a and b as the plane where x0y is located , and use any one of the vectors as The x-axis, the vector a is used as the x-axis in the figure below , as shown in the figure

Then the components of the vector a on the y-axis and the z-axis are both 0, that is $a_{x}=a,a_{y}=0,a_{z}=0$

$a\cdot b=a_{x}b_{x}+a_{y}b_{y}+a_{z}b_{z}=a_{x}b_{x}+0+0=\begin{Vmatrix} a \end{Vmatrix}\begin{Vmatrix} b \end{Vmatrix}cos\theta$ , which $b_{x}$ is the component of the vector b on the x-axis, which is $\begin{Vmatrix} b \end{Vmatrix}cos\theta$

Geometrically, the consistency of the two definitions of vector inner product is verified.

4. Physical interpretation of vector inner product

Assuming that on a slope, the force F moves the weight along the slope to do work, and the displacement is S, as shown in the figure above, the overall work at this time is

$W=F_{x}S_{x}+F_{y}S_{y}$

From another point of view, we decompose the force F along the direction of displacement S and the direction perpendicular to S, and the work done is

$W=F_{s}S=FScos\theta$

The work of the two is consistent, so we have verified the consistency of the two definitions of the vector inner product from a physical point of view.

Study books:

"The Geometric Meaning of Linear Algebra" Ren Guangqian

3D Mathematical Fundamentals - Graphics and Game Development (2nd Edition) by Fletcher Dunn

Vector Inner Product and Its Interpretation

1. Vector

2. Definition of inner product of vector

3. Geometric interpretation of vector inner product

4. Physical interpretation of vector inner product

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