Coordinate Systems in Space, Space Vectors - Analytic Geometry and Vectors

University Mathematics Handbook (2015)

III. Analytic Geometry and Vectors

Chapter 4. Coordinate Systems in Space, Space Vectors

4.1 Vector Concept

a. A vector is a defined quantity with both direction and magnitude.

b. It is graphically represented by a directional segment, that is, a segment with an arrow on one end, representing the direction of the vector.

c. The absolute value of a vector is called the length or a vector.

d. Vectors are denoted by small, bold Latin characters , and modules are denoted as . A vector starting at point and ending at point is denoted as and its length is denoted as or .

e. is the opposite vector of so they are denoted as .

f. A vector of length is called unit vector.

g. Two vectors are equal if their lengths are equal and their directions are identical.

h. The angle between vectors is the angle between their positive-direction parts. So, if vectors and are parallel to each other, that is, they are situated on parallel lines, and have the same direction, the angle between them is zero. If vectors and are parallel to each other and are of opposite directions, the angle between them is.

4.2 Vector Algebra

a. Vector additions: vector , the sum of vectors and , is denoted , and defined the following way: displace vectors and to common initial point . Vector is the diagonal of parallelogram constructed in vectors and from to .

Vector can also be constructed this way: Draw ; from the end of , draw . Connecting the initial point of to the end of , we obtain vector .

b. Adding more than two vectors is defined similarly.

c. If is the angle between vectors and , then:

d. Vector subtraction: The difference of vectors and is vector , for which holds .

e. The projection of vector on is scalar which equals to the length of directed segment , when is the projection of the initial point and , the projection of the terminal point of , marked with plus when is at the direction of , and with minus when it is in the opposite direction of .

f. The projection of the sum of two vectors equals to the sum of their projections.

g. The projection of a difference of two vectors equals to the difference of their projections.

4.3 Vector Triangle Inequality

For all two vectors a and b there holds

4.4 Cartesian Coordinate System in Space

a. Unit vectors define a right-handed Cartesian system, if:

1. They are perpendicular to each other

2. All three vectors are initiated in one point, and, the motion from to counterclockwise is visible from the terminal point of vector .

b. Cartesian coordinate system in space consists of three axes originating from one point : -axis at the direction of vector ; -axis at the direction of vector , and -axis at the direction of vector .

c. Every vector can be represented in the form of . Three entities are called Cartesian coordinates.

d. The space coordinates of point are the space coordinates of vector .

e. For every point in space there is a corresponding ordered set of three real numbers, and for every ordered set of three real numbers, there is a corresponding point in space.

f. Example: In the following Figure, there are points

4.5 Vector in Coordinate System

Let's give vectors ,, and scalar , then:

e. is parallel to if, and only if,

4.6 Vector Direction in Space

a. Direction angles of vector , are angles it forms with the positive direction of axes , respectively.

b. Direction cosines of are .

c. If , then

4.7 Inner (Scalar) Product

a. The inner product of two vectors , is scalar ( dot ), defined as:

when is the angle between the vectors.

b. , when is the projection of on , and is the projection of on .

c. Vectors and are mutually perpendicular if, and only if, .

d. If and , then:

e. , ( is a scalar).

f. .

g. .

4.8 Cross (Vector) Product

a. Cross product of vectors , is vector denoted as ( cross ), which holds:

1. , when is the angle between vectors , . In other words, the length of vector equals the area of the parallelogram with vectors for sides.

2. Vector is perpendicular to the plane of vectors and , and its direction is such that, from the terminal point of the motion from to counterclockwise is visible (right-hand rule).