[TA frost wolf \u may- hundred people plan] 1.2.1 vector basis

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1.2.1.1 vector

The definition of vector

• A vector is a directed line segment with size and direction
• The vector has no position
• The arrow of the vector is the end of the vector , The tail of the vector is the beginning of the vector
• The displacement described by the vector can be considered as a displacement sequence parallel to the axis
• Representation of vectors ： The three dimensional （ax,ay,az） for example （1,-5,7）

Vector and scalar

vector ： There is size and direction
Scalar ： Only size, no direction

Vectors and points

The mathematical form of a vector is the same as that of a point , But the geometric meaning is completely different
spot ： Only location information
vector ： No location information , But there is actual size and orientation information

contact ： Any point can be regarded as a vector starting from the origin

Zero vector

• The zero vector is the only vector of zero size
• The zero vector is the only quantity that has no direction
• The zero vector is not a point , Because the zero vector doesn't define a position
• Zero vector means that there is no unique , Just like zero scalar means no quantity

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1.2.1.2 Calculation

Calculation of scalar and vector

• No addition
• No subtraction
• Multiplication ： Multiply each component of the vector by the scalar
• division ： Equivalent to multiplying the reciprocal of scalar

The module length of the vector

• Calculation formula ：||v|| = √vx²+vy²
• Geometric interpretation ： Construct a right triangle with a vector as the hypotenuse , The size of the vector shown （ Die length ） It can be obtained from Pythagorean theorem

Normalized vectors

• Normalized vectors （ Unit vector ） The module length is 1 Vector .（ Just know the direction , application ： normal ）
• Algorithm ： Divide the components of the vector by the module length

Addition and subtraction of vectors

• Calculation formula ：（ax,ay）±（bx,by） = （ax+bx,ay+by）
• The addition and subtraction of corresponding positions
• Geometric interpretation ： Superposition of displacements in all directions

Calculate the distance between two points

• Calculation formula ：（a,b）= ||b-a|| = √（bx-ax）²+（by-ay）²
• And so on in higher space
• Range of application ： Calculate the distance from one vector to another （a To b The displacement vector of is b-a）

Dot product operation of vector

• Calculation formula ：（ax,ay）·（bx,by）=（axbx+ayby）
• Vector point multiplication is the sum of component products , Satisfying the commutative law
• Geometric interpretation ： The result of point multiplication describes the of two vectors “ be similar ” Degree of , The greater the result of dot multiplication , The smaller the angle , The closer the two vectors are .

Projection

• The projection length of one vector on another vector

Lambert illumination model

Lambert illumination model is the simplest illumination model used to simulate diffuse reflection .
set up ： The opposite direction of the illumination direction is L vector , The discovery direction is N vector , Then there are ：

• L And N In the same direction ：Nor·Light = 1（ Pure bright ）
• L And N When the direction is opposite ：Nor·Light = -1（ Pure dark ）
• L And N When the direction is vertical ：Nor·Light = 0（ Pure dark ）

The cross product of vectors

Only for 3D vector , The calculation formula is shown in the figure ：

Does not satisfy the law of exchange , But it satisfies the inverse commutative law ：
a X b ≠ b X a
a X b = - (b X a)

• Vector cross multiplication is the cross multiplication and subtraction of components , The result is a vector .
• Geometric interpretation ： The vector obtained by cross multiplication is perpendicular to the original two vectors .

Determine the size and direction of cross product

The calculation formula is shown in the figure ：

The direction follows the left-handed coordinate system , When the starting points coincide , from a The vector turns b vector , The direction of the thumb is the direction of the result .