Vectors and parallelism

The triangle rule (Chasles' relation)

For any three points in the plane,

the following equality holds:

The parallelogram rule

Let V and

Construct the parallelogram

Then

Subtracting two vectors

Subtracting two vectors means adding the first vector to the opposite of the second.

Collinear vectors

Let be a non-zero vector and an arbitrary vector.

1. If and are collinear, then there is a unique real number such that .

2. If there exists such that , then and are collinear.

Let and be two non-collinear vectors. For any vector in V there exist such that . The scalars and with this property are unique.

Theorem:

1. For any and , (commutativity)

2. For any , , , (associativity)

3. For any , ( is the identity element)

4. For any , (every vector has an opposite).

Theorem: For any and any , we have:

1. 
2. 
3. 
4. 

Definition: Let be a rectangular coordinate system in the plane and the points A(1,0) and B(0,1). We write the vectors , ; are called the unit vectors of the coordinate axes of the system . The pair () is called the basis of the system .

Definition: Let be a rectangular coordinate system in the plane. For any point M in the plane, the vector is called the position vector of the point M.

Definition: In space we fix three axes , , with the same origin , pairwise perpendicular. The resulting system is written and is called a three-dimensional rectangular (or Cartesian) coordinate system. The orientation of the axes is usually chosen as the arrows in the adjacent figure show.

Definition: In the system , each point M(x,z,y) in space is determined by the position vector ; are called the coordinates of the vector in the basis (); x is called the abscissa, y the ordinate, and z the applicate of the point M(x,y,z).

Definition: The norm of a vector is the distance between its endpoints.

Let the points and . Then .

The scalar product. For any two vectors , the real number where is called the scalar product of the vectors . If is zero, then by definition the scalar product is zero.

Properties of the scalar product.

Let

1. (commutativity)

2. ( is the scalar projection of onto )

3. (distributivity)

4. (moving the scalar)

5. If and , then

6. The scalar product is zero if and only if or or .

Theorem. Let and be vectors, the points , , and let be the angle between the vectors and . Then

Definition. Let be the position vector of the point M. The angles which the direction (line, vector) makes with the positive directions of the coordinate axes , , are called direction angles. The cosines of these angles are called direction cosines.

Theorem. Let be the position vector of the point . The vector makes with the axes the direction angles with , , .

If are the direction cosines of the line d and , then is the normal equation of the plane perpendicular to the line d, at distance p from the origin.

The equation of the line passing through the point with direction vector is: .

The equation of the line passing through the points , is: .

The distinct lines and are parallel if and only if , and perpendicular if and only if .