Knowledge and understanding of mathematical terms will help in solving many tasks as a course of algebra and geometry. An equally important role is given to formulas that displays the relationship between mathematical characteristics.
The angle between vectors - explanation of terminology
In order to formulate the definition of the angle between vectors, it is necessary to find out what implies the term "vector". This concept characterizes a straight line, which has the beginning, length and direction. If you are depicted 2 directed segments that originate in the same point, therefore they form an angle.
That. The term "the angle between vectors" determines the degree of the smallest angle to which one directional segment should be turned (relative to the starting point) so that it takes the position / direction of the second directional portion. This statement applies to the vector vectors from one point.
The degree of the corner between the two directed areas of the straight, originated at one point is concluded in the segment from 0 º up to 180. º. This value is designated as ∠ (ā, ū) - the angle between the directed segments ā and ū.
Calculation of the corner between vectors
The calculation of the degree measure of the angle formed by a pair of directed parts of the line is made using the following formula:
cosφ \u003d (ō, ā) / | ō | · ā |, ⇒ φ \u003d arccos (cosφ).
∠φ - the desired angle between the specified vectors ō and ā,
(ō, ā) - the work of the regiments of the directed parts of the line,
| ō | · | ā | - The product of the lengths of the given directed segments.
Determination of a scalar product of directed areas
How to use this formula and determine the value of the numerator and denominator of the presented relationship?
Depending on the coordinate system (decartian or three-dimensional space), in which the specified vectors are located, each directional segment has the following parameters:
ō = { o.x., o.y.}, ā = { a. x., a.y.) or
ō = { o.x., o.y.O.z.}, ā = { a. x., a.y., A.z.}.
Consequently, to find the value of the numerator - the scalalar of the directed segments - such actions should be made:
(ō,ā) = ō * ā = o.x.* a. x.+ o.y.* A.y.if the vector under consideration lie on the plane
(ō,ā) = ō * ā = o.x.* a. x.+ o.y.* A.y.+ o.z.* a.z.If the directed areas are located in space.
Determination of vectors
The length of the directional segment is calculated using expressions:
|ō| = √ o.x.2.+ o.y.2.or | ō | \u003d √ o.x.2.+ o.y.2.+ o.z.2
| ā | \u003d √ A. x.2.+ a.y.2.or | ā | \u003d √ a.x.2.+ a.y.2.+ a.z.2
That. In the general case of n-dimensional measurement, the expression to determine the degree of the angle between the directed segments ō \u003d ( o.x., o.y.... O.n.) and ā \u003d ( a. x., a.y.... A.n.) looks like that:
φ \u003d arccos (cosφ) \u003d ArcCOS (( o.x.* a. x.+ o.y.* A.y.+ … + o.n.* a.n.) / (√ o.x.2.+ o.y.2.+ … + o.n.2 * √ a.x.2.+ a.y.2.+ … + a.n.2) ).
An example of calculating the angle between directional segments
According to the conditions, the vectors ī \u003d (3; 4; 0) and ū \u003d (4; 4; 2) are given. What is the degree of a measure of an angle formed by these segments?
Determine the scalar of vectors ī and ū. For this:
i * U \u003d 3 * 4 + 4 * 4 + 0 * 2 \u003d 28
After calculating the length of the segments:
| ī | \u003d √9 + 16 + 0 \u003d √25 \u003d 5,
| ū ū | \u003d √16 + 16 + 4 \u003d √36 \u003d 6.
cOS (ī, ū) \u003d 28/5 * 6 \u003d 28/30 \u003d 14/15 \u003d 0.9 (3).
Taking advantage of the table of cosine (bradys) values, determine the magnitude of the original angle:
cOS (ī, ū) \u003d 0.9 (3) ⇒ ∠ (ī, ū) \u003d 21 ° 6 '.
348
