Definition
Scalar quantity- a quantity that can be characterized by a number. For example, length, area, mass, temperature, etc.
Vector called the directed segment $\overline(A B)$; point $A$ is the beginning, point $B$ is the end of the vector (Fig. 1).
A vector is denoted either by two capital letters - its beginning and end: $\overline(A B)$ or by one small letter: $\overline(a)$.
Definition
If the beginning and end of a vector coincide, then such a vector is called zero. Most often, the zero vector is denoted as $\overline(0)$.
The vectors are called collinear, if they lie either on the same line or on parallel lines (Fig. 2).
Definition
Two collinear vectors $\overline(a)$ and $\overline(b)$ are called co-directed, if their directions coincide: $\overline(a) \uparrow \uparrow \overline(b)$ (Fig. 3, a). Two collinear vectors $\overline(a)$ and $\overline(b)$ are called oppositely directed, if their directions are opposite: $\overline(a) \uparrow \downarrow \overline(b)$ (Fig. 3, b).
Definition
The vectors are called coplanar, if they are parallel to the same plane or lie in the same plane (Fig. 4).
Two vectors are always coplanar.
Definition
Length (module) vector $\overline(A B)$ is the distance between its beginning and end: $|\overline(A B)|$
Detailed theory about vector length at the link.
The length of the zero vector is zero.
Definition
A vector whose length is equal to one is called unit vector or ortom.
The vectors are called equal, if they lie on one or parallel lines; their directions coincide and their lengths are equal.
There will also be problems for you to solve on your own, to which you can see the answers.
Vector concept
Before you learn everything about vectors and operations on them, get ready to solve a simple problem. There is a vector of your entrepreneurship and a vector of your innovative abilities. The vector of entrepreneurship leads you to Goal 1, and the vector of innovative abilities leads you to Goal 2. The rules of the game are such that you cannot move along the directions of these two vectors at once and achieve two goals at once. Vectors interact, or, speaking in mathematical language, some operation is performed on vectors. The result of this operation is the “Result” vector, which leads you to Goal 3.
Now tell me: the result of which operation on the vectors “Entrepreneurship” and “Innovative abilities” is the vector “Result”? If you can't tell right away, don't be discouraged. As you progress through this lesson, you will be able to answer this question.
As we have already seen above, the vector necessarily comes from a certain point A in a straight line to some point B. Consequently, each vector has not only a numerical value - length, but also a physical and geometric value - direction. From this comes the first, simplest definition of a vector. So, a vector is a directed segment coming from a point A to the point B. It is designated as follows: .
And to begin various operations with vectors , we need to get acquainted with one more definition of a vector.
A vector is a type of representation of a point that needs to be reached from some starting point. For example, a three-dimensional vector is usually written as (x, y, z) . In very simple terms, these numbers mean how far you need to walk in three different directions to get to a point.
Let a vector be given. At the same time x = 3 (right hand points to the right), y = 1 (left hand points forward) z = 5 (under the point there is a staircase leading up). Using this data, you will find a point by walking 3 meters in the direction indicated by your right hand, then 1 meter in the direction indicated by your left hand, and then a ladder awaits you and, rising 5 meters, you will finally find yourself at the end point.
All other terms are clarifications of the explanation presented above, necessary for various operations on vectors, that is, solving practical problems. Let's go through these more rigorous definitions, focusing on typical vector problems.
Physical examples vector quantities can be the displacement of a material point moving in space, the speed and acceleration of this point, as well as the force acting on it.
Geometric vector presented in two-dimensional and three-dimensional space in the form directional segment. This is a segment that has a beginning and an end.
If A- the beginning of the vector, and B- its end, then the vector is denoted by the symbol or one lowercase letter . In the figure, the end of the vector is indicated by an arrow (Fig. 1)
Length(or module) of a geometric vector is the length of the segment generating it
The two vectors are called equal , if they can be combined (if the directions coincide) by parallel transfer, i.e. if they are parallel, directed in the same direction and have equal lengths.
In physics it is often considered pinned vectors, specified by the point of application, length and direction. If the point of application of the vector does not matter, then it can be transferred, maintaining its length and direction, to any point in space. In this case, the vector is called free. We will agree to consider only free vectors.
Linear operations on geometric vectors
Multiplying a vector by a number
Product of a vector per number is a vector that is obtained from a vector by stretching (at ) or compressing (at ) by a factor, and the direction of the vector remains the same if , and changes to the opposite if . (Fig. 2)
From the definition it follows that the vectors and = are always located on one or parallel lines. Such vectors are called collinear. (We can also say that these vectors are parallel, but in vector algebra it is customary to say “collinear.”) The converse is also true: if the vectors are collinear, then they are related by the relation
Consequently, equality (1) expresses the condition of collinearity of two vectors.
Addition and subtraction of vectors
When adding vectors you need to know that amount vectors and is called a vector, the beginning of which coincides with the beginning of the vector, and the end - with the end of the vector, provided that the beginning of the vector is attached to the end of the vector. (Fig. 3)
This definition can be distributed over any finite number of vectors. Let them be given in space n free vectors. When adding several vectors, their sum is taken to be the closing vector, the beginning of which coincides with the beginning of the first vector, and the end with the end of the last vector. That is, if you attach the beginning of the vector to the end of the vector, and the beginning of the vector to the end of the vector, etc. and, finally, to the end of the vector - the beginning of the vector, then the sum of these vectors is the closing vector 
, the beginning of which coincides with the beginning of the first vector, and the end - with the end of the last vector. (Fig. 4)
The terms are called components of the vector, and the formulated rule is polygon rule. This polygon may not be flat.
When a vector is multiplied by the number -1, the opposite vector is obtained. The vectors and have the same lengths and opposite directions. Their sum gives zero vector, whose length is zero. The direction of the zero vector is not defined.
In vector algebra, there is no need to consider the subtraction operation separately: subtracting a vector from a vector means adding the opposite vector to the vector, i.e. 
Example 1. Simplify the expression:
.
,
that is, vectors can be added and multiplied by numbers in the same way as polynomials (in particular, also problems on simplifying expressions). Typically, the need to simplify linearly similar expressions with vectors arises before calculating the products of vectors.
Example 2. Vectors and serve as diagonals of the parallelogram ABCD (Fig. 4a). Express through and the vectors , , and , which are the sides of this parallelogram.
Solution. The point of intersection of the diagonals of a parallelogram bisects each diagonal. We find the lengths of the vectors required in the problem statement either as half the sums of the vectors that form a triangle with the required ones, or as half the differences (depending on the direction of the vector serving as the diagonal), or, as in the latter case, half the sum taken with a minus sign. The result is the vectors required in the problem statement:
There is every reason to believe that you have now correctly answered the question about the vectors “Entrepreneurship” and “Innovative abilities” at the beginning of this lesson. Correct answer: an addition operation is performed on these vectors.
Solve vector problems yourself and then look at the solutions
How to find the length of the sum of vectors?
This problem occupies a special place in operations with vectors, since it involves the use of trigonometric properties. Let's say you come across a task like the following:
The vector lengths are given 
and the length of the sum of these vectors. Find the length of the difference between these vectors.
Solutions to this and other similar problems and explanations of how to solve them are in the lesson " Vector addition: length of the sum of vectors and the cosine theorem ".
And you can check the solution to such problems at Online calculator "Unknown side of a triangle (vector addition and cosine theorem)" .
Where are the products of vectors?
Vector-vector products are not linear operations and are considered separately. And we have lessons "Scalar product of vectors" and "Vector and mixed products of vectors".
Projection of a vector onto an axis
The projection of a vector onto an axis is equal to the product of the length of the projected vector and the cosine of the angle between the vector and the axis:
As is known, the projection of a point A on the straight line (plane) is the base of the perpendicular dropped from this point onto the straight line (plane).
Let be an arbitrary vector (Fig. 5), and and be the projections of its origin (points A) and end (points B) per axis l. (To construct a projection of a point A) draw a straight line through the point A a plane perpendicular to a straight line. The intersection of the line and the plane will determine the required projection.
Vector component on the l axis is called such a vector lying on this axis, the beginning of which coincides with the projection of the beginning, and the end with the projection of the end of the vector.
Projection of the vector onto the axis l called number
,
equal to the length of the component vector on this axis, taken with a plus sign if the direction of the components coincides with the direction of the axis l, and with a minus sign if these directions are opposite.
Basic properties of vector projections onto an axis:
1. Projections of equal vectors onto the same axis are equal to each other.
2. When a vector is multiplied by a number, its projection is multiplied by the same number.
3. The projection of the sum of vectors onto any axis is equal to the sum of the projections of the summands of the vectors onto the same axis.
4. The projection of the vector onto the axis is equal to the product of the length of the projected vector and the cosine of the angle between the vector and the axis:
.
Solution. Let's project vectors onto the axis l as defined in the theoretical background above. From Fig. 5a it is obvious that the projection of the sum of vectors is equal to the sum of the projections of vectors. We calculate these projections:
We find the final projection of the sum of vectors:
Relationship between a vector and a rectangular Cartesian coordinate system in space
Getting to know rectangular Cartesian coordinate system in space took place in the corresponding lesson, it is advisable to open it in a new window.
In an ordered system of coordinate axes 0xyz axis Ox called x-axis, axis 0y – y-axis, and axis 0z – axis applicate.
With an arbitrary point M space connect vector
called radius vector points M and project it onto each of the coordinate axes. Let us denote the magnitudes of the corresponding projections:
Numbers x, y, z are called coordinates of point M, respectively abscissa, ordinate And applicate, and are written as an ordered point of numbers: M(x;y;z)(Fig. 6).
A vector of unit length whose direction coincides with the direction of the axis is called unit vector(or ortom) axes. Let us denote by
Accordingly, the unit vectors of the coordinate axes Ox, Oy, Oz
Theorem. Any vector can be expanded into unit vectors of coordinate axes:
(2)
Equality (2) is called the expansion of the vector along the coordinate axes. The coefficients of this expansion are the projections of the vector onto the coordinate axes. Thus, the coefficients of expansion (2) of the vector along the coordinate axes are the coordinates of the vector.
After choosing a certain coordinate system in space, the vector and the triplet of its coordinates uniquely determine each other, so the vector can be written in the form
Representations of the vector in the form (2) and (3) are identical.
Condition for collinearity of vectors in coordinates
As we have already noted, vectors are called collinear if they are related by the relation
Let the vectors be given 
. These vectors are collinear if the coordinates of the vectors are related by the relation
,
that is, the coordinates of the vectors are proportional.
Example 6. Vectors are given 
. Are these vectors collinear?
Solution. Let's find out the relationship between the coordinates of these vectors:
.
The coordinates of the vectors are proportional, therefore, the vectors are collinear, or, what is the same, parallel.
Vector length and direction cosines
Due to the mutual perpendicularity of the coordinate axes, the length of the vector
equal to the length of the diagonal of a rectangular parallelepiped built on vectors
and is expressed by the equality
(4)
A vector is completely defined by specifying two points (start and end), so the coordinates of the vector can be expressed in terms of the coordinates of these points.
Let, in a given coordinate system, the origin of the vector be at the point
and the end is at the point
From equality
It follows that
or in coordinate form
Hence, vector coordinates are equal to the differences between the same coordinates of the end and beginning of the vector . Formula (4) in this case will take the form
The direction of the vector is determined direction cosines . These are the cosines of the angles that the vector makes with the axes Ox, Oy And Oz. Let us denote these angles accordingly α , β And γ . Then the cosines of these angles can be found using the formulas
The direction cosines of a vector are also the coordinates of the vector of that vector and thus the vector of the vector
.
Considering that the length of the unit vector is equal to one unit, that is
,
we obtain the following equality for the direction cosines:
Example 7. Find the length of the vector x = (3; 0; 4).
Solution. The length of the vector is
Example 8. Points given:
Find out whether the triangle constructed on these points is isosceles.
Solution. Using the vector length formula (6), we find the lengths of the sides and determine whether there are two equal ones among them:
Two equal sides have been found, therefore there is no need to look for the length of the third side, and the given triangle is isosceles.
Example 9. Find the length of the vector and its direction cosines if 
.
Solution. The vector coordinates are given:
.
The length of the vector is equal to the square root of the sum of the squares of the vector coordinates:
.
Finding direction cosines:
Solve the vector problem yourself, and then look at the solution
Operations on vectors given in coordinate form
Let two vectors and be given, defined by their projections:
Let us indicate actions on these vectors.
The length of the vector a → will be denoted by a → . This notation is similar to the modulus of a number, so the length of a vector is also called the modulus of a vector.
To find the length of a vector on a plane from its coordinates, it is necessary to consider a rectangular Cartesian coordinate system O x y. Let some vector a → with coordinates a x be specified in it; ay. Let us introduce a formula for finding the length (modulus) of the vector a → through the coordinates a x and a y.
Let us plot the vector O A → = a → from the origin. Let us define the corresponding projections of point A onto the coordinate axes as A x and A y. Now consider a rectangle O A x A A y with diagonal O A .
From the Pythagorean theorem follows the equality O A 2 = O A x 2 + O A y 2 , whence O A = O A x 2 + O A y 2 . From the already known definition of vector coordinates in a rectangular Cartesian coordinate system, we obtain that O A x 2 = a x 2 and O A y 2 = a y 2 , and by construction, the length of O A is equal to the length of the vector O A → , which means O A → = O A x 2 + O A y 2.
From this it turns out that formula for finding the length of a vector a → = a x ; a y has the corresponding form: a → = a x 2 + a y 2 .
If the vector a → is given in the form of an expansion in coordinate vectors a → = a x i → + a y j →, then its length can be calculated using the same formula a → = a x 2 + a y 2, in this case the coefficients a x and a y are as the coordinates of the vector a → in a given coordinate system.
Example 1
Calculate the length of the vector a → = 7 ; e, specified in a rectangular coordinate system.
Solution
To find the length of a vector, we will use the formula for finding the length of a vector from coordinates a → = a x 2 + a y 2: a → = 7 2 + e 2 = 49 + e
Answer: a → = 49 + e.
Formula for finding the length of a vector a → = a x ; a y ; a z from its coordinates in the Cartesian coordinate system Oxyz in space, is derived similarly to the formula for the case on a plane (see figure below)
In this case, O A 2 = O A x 2 + O A y 2 + O A z 2 (since OA is the diagonal of a rectangular parallelepiped), hence O A = O A x 2 + O A y 2 + O A z 2 . From the definition of vector coordinates we can write the following equalities O A x = a x ; O A y = a y ; O A z = a z ; , and the length OA is equal to the length of the vector that we are looking for, therefore, O A → = O A x 2 + O A y 2 + O A z 2 .
It follows that the length of the vector a → = a x ; a y ; a z is equal to a → = a x 2 + a y 2 + a z 2 .
Example 2
Calculate the length of the vector a → = 4 · i → - 3 · j → + 5 · k → , where i → , j → , k → are the unit vectors of the rectangular coordinate system.
Solution
The vector decomposition a → = 4 · i → - 3 · j → + 5 · k → is given, its coordinates are a → = 4, - 3, 5. Using the above formula we get a → = a x 2 + a y 2 + a z 2 = 4 2 + (- 3) 2 + 5 2 = 5 2.
Answer: a → = 5 2 .
Length of a vector through the coordinates of its start and end points
Formulas were derived above that allow you to find the length of a vector from its coordinates. We considered cases on a plane and in three-dimensional space. Let's use them to find the coordinates of a vector from the coordinates of its start and end points.
So, points with given coordinates A (a x ; a y) and B (b x ; b y) are given, hence the vector A B → has coordinates (b x - a x ; b y - a y) which means its length can be determined by the formula: A B → = ( b x - a x) 2 + (b y - a y) 2
And if points with given coordinates A (a x ; a y ; a z) and B (b x ; b y ; b z) are given in three-dimensional space, then the length of the vector A B → can be calculated using the formula
A B → = (b x - a x) 2 + (b y - a y) 2 + (b z - a z) 2
Example 3
Find the length of the vector A B → if in the rectangular coordinate system A 1, 3, B - 3, 1.
Solution
Using the formula for finding the length of a vector from the coordinates of the start and end points on the plane, we obtain A B → = (b x - a x) 2 + (b y - a y) 2: A B → = (- 3 - 1) 2 + (1 - 3) 2 = 20 - 2 3 .
The second solution involves applying these formulas in turn: A B → = (- 3 - 1 ; 1 - 3) = (- 4 ; 1 - 3) ; A B → = (- 4) 2 + (1 - 3) 2 = 20 - 2 3 . -
Answer: A B → = 20 - 2 3 .
Example 4
Determine at what values the length of the vector A B → is equal to 30 if A (0, 1, 2); B (5 , 2 , λ 2) .
Solution
First, let's write down the length of the vector A B → using the formula: A B → = (b x - a x) 2 + (b y - a y) 2 + (b z - a z) 2 = (5 - 0) 2 + (2 - 1) 2 + (λ 2 - 2) 2 = 26 + (λ 2 - 2) 2
Then we equate the resulting expression to 30, from here we find the required λ:
26 + (λ 2 - 2) 2 = 30 26 + (λ 2 - 2) 2 = 30 (λ 2 - 2) 2 = 4 λ 2 - 2 = 2 and λ 2 - 2 = - 2 λ 1 = - 2, λ 2 = 2, λ 3 = 0.
Answer: λ 1 = - 2, λ 2 = 2, λ 3 = 0.
Finding the length of a vector using the cosine theorem
Alas, in problems the coordinates of the vector are not always known, so we will consider other ways to find the length of the vector.
Let the lengths of two vectors A B → , A C → and the angle between them (or the cosine of the angle) be given, and you need to find the length of the vector B C → or C B → . In this case, you should use the cosine theorem in the triangle △ A B C and calculate the length of the side B C, which is equal to the desired length of the vector.
Let's consider this case using the following example.
Example 5
The lengths of the vectors A B → and A C → are 3 and 7, respectively, and the angle between them is π 3. Calculate the length of the vector B C → .
Solution
The length of the vector B C → in this case is equal to the length of the side B C of the triangle △ A B C . The lengths of the sides A B and A C of the triangle are known from the condition (they are equal to the lengths of the corresponding vectors), the angle between them is also known, so we can use the cosine theorem: B C 2 = A B 2 + A C 2 - 2 A B A C cos ∠ (A B, → A C →) = 3 2 + 7 2 - 2 · 3 · 7 · cos π 3 = 37 ⇒ B C = 37 Thus, B C → = 37 .
Answer: B C → = 37 .
So, to find the length of a vector from coordinates, there are the following formulas a → = a x 2 + a y 2 or a → = a x 2 + a y 2 + a z 2 , from the coordinates of the start and end points of the vector A B → = (b x - a x) 2 + ( b y - a y) 2 or A B → = (b x - a x) 2 + (b y - a y) 2 + (b z - a z) 2, in some cases the cosine theorem should be used.
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Oxy
ABOUT A OA.
, where 
OA 
.
Thus, 
.
Let's look at an example.
Example.
Solution.
:
Answer:
Oxyz in space.
A OA will be a diagonal.
In this case (since OA 
OA 
.
Thus, vector length 
.
Example.
Calculate Vector Length 
Solution.
, hence, 
Answer:
Straight line on a plane
General equation
Ax + By + C ( > 0).
Vector = (A; B) is a normal vector.
In vector form: + C = 0, where is the radius vector of an arbitrary point on a line (Fig. 4.11).
Special cases:
1) By + C = 0- straight line parallel to the axis Ox;
2) Ax + C = 0- straight line parallel to the axis Oy;
3) Ax + By = 0- the straight line passes through the origin;
4) y = 0- axis Ox;
5) x = 0- axis Oy.
Equation of a line in segments
Where a, b- the values of the segments cut off by the straight line on the coordinate axes.
Normal equation of a line(Fig. 4.11)
where is the angle formed normal to the line and the axis Ox; p- the distance from the origin to the straight line.
Reducing the general equation of a line to normal form:
Here is the normalized factor of the line; the sign is chosen opposite to the sign C, if and arbitrarily, if C=0.
Finding the length of a vector from coordinates.
We will denote the length of the vector by . Because of this notation, the length of a vector is often called the modulus of the vector.
Let's start by finding the length of a vector on a plane using coordinates.
Let us introduce a rectangular Cartesian coordinate system on the plane Oxy. Let a vector be specified in it and have coordinates . We obtain a formula that allows us to find the length of a vector through the coordinates and .
Let us postpone from the origin of coordinates (from the point ABOUT) vector . Let us denote the projections of the point A on the coordinate axes as and respectively and consider a rectangle with a diagonal OA.
By virtue of the Pythagorean theorem, the equality , where . From the definition of vector coordinates in a rectangular coordinate system, we can state that and , and by construction the length OA equal to the length of the vector, therefore, .
Thus, formula for finding the length of a vector according to its coordinates on the plane has the form .
If the vector is represented as a decomposition in coordinate vectors , then its length is calculated using the same formula , since in this case the coefficients and are the coordinates of the vector in a given coordinate system.
Let's look at an example.
Example.
Find the length of the vector given in the Cartesian coordinate system.
Solution.
We immediately apply the formula to find the length of the vector from the coordinates :
Answer:
Now we get the formula for finding the length of the vector by its coordinates in a rectangular coordinate system Oxyz in space.
Let us plot the vector from the origin and denote the projections of the point A on the coordinate axes as and . Then we can construct a rectangular parallelepiped on the sides, in which OA will be a diagonal.
In this case (since OA– diagonal of a rectangular parallelepiped), from where . Determining the coordinates of a vector allows us to write equalities, and the length OA equal to the desired vector length, therefore, .
Thus, vector length in space is equal to the square root of the sum of the squares of its coordinates, that is, found by the formula .
Example.
Calculate Vector Length , where are the unit vectors of the rectangular coordinate system.
Solution.
We are given a vector decomposition into coordinate vectors of the form , hence, . Then, using the formula for finding the length of a vector from coordinates, we have .
Before moving on to the topic of the article, let us recall the basic concepts.
Definition 1
Vector– a straight line segment characterized by a numerical value and direction. A vector is denoted by a lowercase Latin letter with an arrow on top. If there are specific boundary points, the vector designation looks like two capital Latin letters (marking the boundaries of the vector) also with an arrow on top.
Definition 2
Zero vector– any point on the plane, designated as zero with an arrow on top.
Definition 3
Vector length– a value equal to or greater than zero that determines the length of the segment that makes up the vector.
Definition 4
Collinear vectors– lying on one line or on parallel lines. Vectors that do not fulfill this condition are called non-collinear.
Definition 5Input: vectors a → And b →. To perform an addition operation on them, it is necessary to plot a vector from an arbitrary undefined point A B →, equal to the vector a →; from the resulting point undefined – vector B C →, equal to the vector b →. By connecting the points undefined and C, we get a segment (vector) A C →, which will be the sum of the original data. Otherwise, the described vector addition scheme is called triangle rule.
Geometrically, vector addition looks like this:
For non-collinear vectors:
For collinear (co-directional or opposite) vectors:
Taking the scheme described above as a basis, we get the opportunity to perform the operation of adding vectors in an amount greater than 2: adding each subsequent vector in turn.
Definition 6
Input: vectors a → , b → , c →, d → . From an arbitrary point A on the plane it is necessary to plot a segment (vector) equal to the vector a →; then from the end of the resulting vector a vector equal to the vector is laid off b →; then, subsequent vectors are laid out using the same principle. The end point of the last deferred vector will be point B, and the resulting segment (vector) A B →– the sum of all initial data. The described scheme for adding several vectors is also called polygon rule .
Geometrically it looks like this:
Definition 7
A separate scheme of action for vector subtraction no, because essentially a vector difference a → And b → is the sum of vectors a → And - b → .
Definition 8To perform the action of multiplying a vector by a certain number k, the following rules must be taken into account:
- if k > 1, then this number will lead to the vector being stretched k times;
- if 0< k < 1 , то это число приведет к сжатию вектора в
1 k times;
- if k< 0 , то это число приведет к смене направления вектора при одновременном выполнении одного из первых двух правил;
- if k = 1, then the vector remains the same;
- if one of the factors is a zero vector or a number equal to zero, the result of the multiplication will be a zero vector.
Initial data:
1) vector a → and number k = 2;
2) vector b → and number k = - 1 3 .
Geometrically, the result of multiplication in accordance with the above rules will look like this:
The operations on vectors described above have properties, some of which are obvious, while others can be justified geometrically.
Input: vectors a → , b → , c → and arbitrary real numbers λ and μ.
The properties of commutativity and associativity make it possible to add vectors in any order.
The listed properties of the operations allow you to carry out the necessary transformations of vector-numeric expressions in a similar way to the usual numeric ones. Let's look at this with an example.
Example 1
Task: simplify the expression a → - 2 · (b → + 3 · a →)
Solution
- using the second distribution property, we get: a → - 2 · (b → + 3 · a →) = a → - 2 · b → - 2 · (3 · a →)
- we use the associative property of multiplication, the expression will take the following form: a → - 2 · b → - 2 · (3 · a →) = a → - 2 · b → - (2 · 3) · a → = a → - 2 · b → - 6 a →
- using the commutativity property, we swap the terms: a → - 2 b → - 6 a → = a → - 6 a → - 2 b →
- then using the first distribution property we get: a → - 6 · a → - 2 · b → = (1 - 6) · a → - 2 · b → = - 5 · a → - 2 · b → A short notation of the solution will look like so: a → - 2 · (b → + 3 · a →) = a → - 2 · b → - 2 · 3 · a → = 5 · a → - 2 · b →
Answer: a → - 2 · (b → + 3 · a →) = - 5 · a → - 2 · b →
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