A vector (Vectors)
Vector a schematic diagram:
- Ordered real numbers to represent different projected length coordinate axes
- There are two expressions of rows and columns in a list of graphics often
- It represents a length in the direction
- Basic operations:
- An adder (associative and commutative)
- Scalar multiplied by a vector (is associative and distribution ratio)
- Dot (dot product, the result is a scalar, the geometric meaning of [formula], and the distribution ratio to satisfy the commutative law, be used to determine the angle between two vectors, find a vector length calculating projection, etc.)
- Cross product (outer product, the result is a vector perpendicular to the plane of the original vector construction, the length of [Formula], two position vector cross product exchange, will be the same length but opposite direction of the vector)
- Vector normalization: the same normalized vector q p p the direction of length 1, can be calculated by multiplying the point
(A) vector normalization
Represents vector length: \ (|| \ || VEC A \)
unit vector:
- Length = 1
- Formula:
\ [\ Hat A = \ {FRAC \ VEC {A} || \ VEC A ||} \] - It may be used to represent a direction
(B) adding a vector
Vector addition diagram:
- Geometry: parallel comply with the law, trigonometry
- Algebraically: adding coordinate
(C) the Cartesian product
Vector addition diagram:
X and Y may be any (usually orthogonal unit) vector,
\(A = {x \choose y }\) \(A^T = \left(x,y \right )\) \(||A|| = \sqrt{x_2+y_2}\)
(Iv) multiply a vector
1. The dot
Vector dot product Illustration:
\(\vec a \cdot \vec b = \left\| \vec a \right\| \left\| \vec b \right\| \cos \theta\) \(\cos \theta = \frac{\vec a \cdot \vec b}{\left\| \vec a \right\| \left\| \vec b \right\|}\)
对于单位向量： \(\cos \theta = \hat a \cdot \hat b\)
(1) the nature of the dot
\ (\ vec a \ times \ vec b = \ vec b \ times \ vec a \)
\ (\ vec a \ times \ left (\ vec b + \ Case C \ right) = \ vec a \ times \ vec b + \ vec a \ times \ vec c \)
\ (\ left (a \ vec a \ right) \ times \ vec b = \ vec a \ times \ left (a \ vec b \ right) = a \ left (\ thing and \ cdot \ vec b \ right) \)
(2) to do the Cartesian product of
a multiplicative combination, and then summed.
- Two-dimensional coordinate system
\ [\ vec a \ cdot \ vec b = {x_a \ choose y_a} \ cdot {x_b \ choose y_b} = x_aX_b + y_ay_b \] - Dimensional coordinate system
\ [\ vec a \ cdot \ vec b = \ begin {pmatrix} x_a \\ y_a \\ z_a \ end {pmatrix} \ cdot \ begin {pmatrix} x_b \\ y_b \\ z_b \ end {pmatrix} = x_ax_b + y_ay_b + z_az_b \]
(3) the dot product projection
schematic:
- \ (\ BOT \ VEC B_ \) : \ (\ VEC B \) in (vec a \ \) \ projected on
- \ (\ vec b_ \ bot \ ) must belong \ (\ VEC A \) (or belonging \ (\ Hat A \) )
- \ (\ Vec b_ \ Shine = a \ vec a \)
- Find the value of k
- \(k = \left\| \vec b_\bot \right\| \cos \theta\)
- \ (\ vec b_ \ bot \ ) must belong \ (\ VEC A \) (or belonging \ (\ Hat A \) )
(4) graphics, a dot
- Sum of two vectors included angle (for example: the cosine of the angle between the surface light source)
- Seeking a projected vector on another vector
Application
direction of the measuring two vectors
1) in a circle
For example, FIG \ (\ vec a \ cdot \ vec b> 0 \) belonging to the forward (Forward), \ (\ VEC A \ CDOT \ VEC C <0 \) after belongs to (Backward)
2) between the two vectors
