Vector is an important geometric object, with the help of its properties it is convenient to solve many problems on the plane and in space. In this article, we give it a definition, consider its main characteristics, and also show how a vector in space can be used to define planes.
What is a vector: two-dimensional case
First of all, you need to clearly understand what kind of object in question. In geometry, a vector is a directed segment. Like any segment, it is characterized by two main elements: the start and end points. The coordinates of these points uniquely determine all the characteristics of the vector.
Consider an example of a vector in the plane. To do this, draw two mutually perpendicular axes x and y. We mark an arbitrary point P (x, y). If we connect this point with the origin (point O), and then indicate the direction to P, then we get the vector OP¯ (later in the article, the line above the symbol indicates that the vector is considered). The vector drawing on the plane is shown below.
Another vector AB¯ is also shown here, and it can be seen that its characteristics are completely identical to OP¯, but it is in another part of the coordinate system. By parallel transfer OP¯, an infinite number of vectors with the same properties can be obtained.
Vector in space
All real objects that surround us are in three-dimensional space. The study of the geometric properties of three-dimensional figures is engaged in stereometry, which operates with the concept of three-dimensional vectors. They differ from two-dimensional only in that their description requires an additional coordinate, which is measured along the third x and y axis perpendicular to the z axis.
The figure below shows a vector in space. The coordinates of its end along each axis are indicated by colored segments. The beginning of the vector is at the intersection of all three coordinate axes, that is, it has coordinates (0; 0; 0).
Since the vector on the plane is a special case of a spatially directed segment, then in the article we will consider only a three-dimensional vector.
The coordinates of the vector according to the known coordinates of its beginning and end
Suppose that there are two points P (x 1 ; y 1 ; z 1 ) and Q (x 2 ; y 2 ; z 2 ). How to determine the coordinates of the vector PQ¯. First, you should agree which of the points will be the beginning and which end of the vector. In mathematics, it is customary to write the object in question along its direction, that is, P is the beginning, Q is the end. Secondly, the coordinates of the vector PQ¯ are calculated as the differences of the corresponding coordinates of the end and the beginning, that is:
PQ¯ = (x 2 - x 1 ; y 2 - y 1 ; z 2 - z 1 ).
Note that by changing the direction of the vector, its coordinates will change sign, like this:
QP¯ = (x 1 - x 2 ; y 1 - y 2 ; z 1 - z 2 ).
This means that PQ¯ = -QP¯.
It is important to understand one more thing. It was said above that in the plane there are countless vectors equal to this one. This fact is also valid for the spatial case. In fact, when we calculated the PQ¯ coordinates in the example above, we carried out the operation of parallel transfer of this vector so that its origin coincides with the origin. The vector PQ¯ can be represented as a directed segment from the origin to the point M ((x 2 - x 1 ; y 2 - y 1 ; z 2 - z 1 ).
Vector Properties
Like any geometry object, a vector has some characteristic characteristics that can be used in solving problems. Briefly list them.
The modulus of a vector is the length of the directional segment. Knowing the coordinates, it is easy to calculate. For the vector PQ¯ in the example above, the module is equal to:
| PQ¯ | = √ [(x 2 - x 1 ) 2 + (y 2 - y 1 ) 2 + (z 2 - z 1 ) 2 ].
The module of the vector on the plane is calculated by a similar formula, only without the participation of the third coordinate.
The sum and difference of the vectors is carried out according to the triangle rule. The figure below shows how the operations of adding and subtracting these objects are performed.
To get the sum vector, you need to attach the beginning of the second to the end of the first vector. The desired vector will begin at the beginning of the first and end at the end of the second vector.
The difference is performed taking into account that the subtracted vector is replaced by the opposite, and then the addition operation described above is carried out.
In addition to addition and subtraction, it is important to be able to multiply the vector by a number. If the number is k, then a vector is obtained whose modulus is k times different from the original one, and the direction either coincides (k> 0) or is opposite to the original one (k <0).
The operation of multiplying vectors among themselves is also defined. For her, we select a separate paragraph in the article.
Scalar and vector multiplication
Suppose that there are two vectors u¯ (x 1 ; y 1 ; z 1 ) and v¯ (x 2 ; y 2 ; z 2 ). Vector by vector can be multiplied in two different ways:
- Scalar. In this case, a number is obtained.
- Vector. The result is some new vector.
The scalar product of the vectors u¯ and v¯ is calculated as follows:
(u¯ * v¯) = | u¯ | * | v¯ | * cos (α).
Where α is the angle between these vectors.
It can be shown that knowing the coordinates u¯ and v¯, their scalar product can be calculated using the following formula:
(u¯ * v¯) = x 1 * x 2 + y 1 * y 2 + z 1 * z 2 .
The scalar product is convenient to use when decomposing a vector into two perpendicularly directed segments. It is also used to calculate the parallelism or orthogonality of vectors, and to calculate the angle between them.
The vector product u¯ and v¯ gives a new vector that is perpendicular to the original one and has a module:
[u¯ * v¯] = | u¯ | * | v¯ | * sin (α).
The up or down direction of the new vector is determined by the rule of the right hand (four fingers of the right hand are directed from the end of the first vector to the end of the second, and the thumb protruding up indicates the direction of the new vector). The figure below shows the result of a vector product for arbitrary a¯ and b¯.
The vector product is used to calculate the area of the figures, as well as when determining the coordinates of a vector perpendicular to a given plane.
Vectors and their properties are conveniently used in determining the plane equation.
Normal and general plane equation
There are several ways to define a plane. One of them is the conclusion of the general equation of the plane, which directly follows from the knowledge of the vector perpendicular to it, and some known point that belongs to the plane.
Suppose that there is a vector n¯ (A; B; C) and a point P (x 0 ; y 0 ; z 0 ). What condition will all the points of the Q (x; y; z) plane satisfy? This condition is the perpendicularity of any vector PQ¯ normal n¯. For two perpendicular vectors, the scalar product becomes equal to zero (cos (90 o ) = 0), we write this:
(n¯ * PQ¯) = 0 or
A * (xx 0 ) + B * (yy 0 ) + C * (zz 0 ) = 0.
Opening the brackets, we get:
A * x + B * y + C * z + (-A * x 0 -B * y 0 -C * z 0 ) = 0 or
A * x + B * y + C * z + D = 0, where D = -A * x 0 -B * y 0 -C * z 0 .
This equation is called common to the plane. We see that the coefficients facing the variables x, y and z are the coordinates of the perpendicular vector n¯. It is called a guide for the plane.
Vector parametric equation of the plane
The second way to determine the plane is to use two vectors lying in it.
Suppose that there are vectors u¯ (x 1 ; y 1 ; z 1 ) and v¯ (x 2 ; y 2 ; z 2 ). As it was said, each of them in space can be represented by an infinite number of identical directed segments, therefore, for a unique definition of the plane, one more point is needed. Let this point be P (x 0 ; y 0 ; z 0 ). Every point Q (x; y; z) will lie in the desired plane if the vector PQ¯ can be represented as a combination of u¯ and v¯. That is, we have:
PQ¯ = α * u¯ + β * v¯.
Where α and β are some real numbers. From this equality follows the expression:
(x; y; z) = (x 0 ; y 0 ; z 0 ) + α * (x 1 ; y 1 ; z 1 ) + β * (x 2 ; y 2 ; z 2 ).
It is called the parametric vector equation of the plane with respect to 2 vectors u¯ and v¯. Substituting arbitrary parameters α and β, we can find all points (x; y; z) that belong to this plane.
From this equation it is easy to obtain a general expression for the plane. To do this, it suffices to find the direction vector n¯ that is perpendicular to both vectors u¯ and v¯, i.e. their vector product should be applied.
The problem of determining the equation of a plane of a general form
We show how to use the above formulas to solve geometric problems. Suppose that the direction vector of the plane is n¯ (5; -3; 1). It is necessary to find the equation of the plane, knowing that the point P (2; 0; 0) belongs to it.
The general equation is written as:
A * x + B * y + C * z + D = 0.
Since the vector perpendicular to the plane is known, the equation will take the form:
5 * x - 3 * y + z + D = 0.
It remains to find the free term D. We calculate it from the knowledge of the coordinates of P:
D = -A * x 0 -B * y 0 -C * z 0 = -5 * 2 + 3 * 0 - 1 * 0 = -10.
Thus, the desired equation of the plane has the form:
5 * x - 3 * y + z -10 = 0.
The figure below shows what the resulting plane is.
The indicated coordinates of the points correspond to the intersections of the plane with the x, y, and z axes.
The problem of determining the plane through two vectors and a point
Now suppose that the previous plane is given differently. Two vectors u¯ (-2; 0; 10) and v¯ (-2; -10/3; 0) are known, as well as the point P (2; 0; 0). How to write the equation of a plane in a vector parametric form? Using the considered corresponding formula, we get:
(x; y; z) = (2; 0; 0) + α * (- 2; 0; 10) + β * (- 2; -10/3; 0).
Note that the definitions of this plane equation, the vectors u¯ and v¯, can be taken absolutely any, but with one condition: they should not be parallel. Otherwise, the plane cannot be unambiguously determined, however, one can find an equation for a beam or a set of planes.