Vector parallel formula

Two vectors a,b are parallel: a=λb (b is not a zero vector); two vectors are perpendicular: the product of the quantities is 0, i.e., a?b=0.

Coordinate representation: a=(x1,y1),b=(x2,y2)

a//b when and only when x1y2-x2y1=0

a⊥b when and only when x1x2+y1y2= 0

In the right-angled coordinate system, we take two unit vectors i and j in the same direction as the x-axis and y-axis, respectively, as bases. Any vector a, from the fundamental theorem of plane vectors, there are and only a pair of real numbers x, y, such that: a = xi + yj, we call (x,y) the (right-angle) coordinates of the vector a, notation: a = (x,y).

Where x is called the coordinate of a on the x-axis, and y is called the coordinate of a on the y-axis, the above equation is called the coordinate representation of the vector. Within a plane rectangular coordinate system, every plane vector can be uniquely represented by a pair of real numbers.

Extension:

If e1 and e2 are two non-***linear in the same plane nonzero vectors, then for any vector a in that plane, there is and only one pair of real numbers λ, μ such that a= λe1+ μe2.

Given the space three vectors a, b, and c, the vector product of vectors a, b, a × b, and then the quantity product (a × b)-c with vector c, the resulting number is called the mixed product of the three vectors a, b, and c, denoted as (a,b,c) or (abc), that is ( abc)=(a,b,c)=(a×b)-c

The mixed product has the following properties:

1. The absolute value of the mixed product of the three non-***-facing vectors a, b, and c is equal to the volume V of the parallel hexahedron with a, b, and c as the prongs and is positive when a, b, and c form a right-handed system; and negative when a, b, and c form a left-handed system. That is, (abc) = εV (ε = 1 when a, b, and c form a right-handed system; ε = -1 when a, b, and c form a left-handed system)

2. Corollary of the above property: the sufficient condition for the three-vector a, b, and c***-plane is (abc) = 0

3. (abc) = (bca) = (cab) = - (bac) = - (cba) = - ( acb)

Reference:

Baidu's Encyclopedia - Planar Vectors