Definition
The addition of vectors is carried out according to triangle rule.
Amount two vectors they call such a third vector, the beginning of which coincides with the beginning, and the end with the end, provided that the end of the vector and the beginning of the vector coincide (Fig. 1).
For addition vectors The parallelogram rule also applies.
Definition
Parallelogram rule- if two non-collinear vectors are brought to a common origin, then the vector coincides with the diagonal of a parallelogram built on vectors (Fig. 2). Moreover, the beginning of the vector coincides with the beginning of the given vectors.
Definition
The vector is called opposite vector to the vector if it collinear vector, equal to it in length, but directed in the opposite direction to the vector.
The vector addition operation has the following properties:
Definition
By difference vectors is called a vector such that the condition is met: (Fig. 3).
Multiplying a vector by a number
Definition
The work vector per number is a vector that satisfies the conditions:
Properties of multiplying a vector by a number:
Here and are arbitrary vectors, and are arbitrary numbers.
Euclidean space(Also Euclidean space) - in the original sense, the space whose properties are described axioms Euclidean geometry. In this case, it is assumed that the space has dimension equal to 3.
In the modern sense, in a more general sense, it can mean one of similar and closely related objects: finite-dimensional real vector space with a positive definite introduced on it scalar product, or metric space, corresponding to such a vector space. In this article, the first definition will be taken as the starting point.
Dimensional Euclidean space is also often denoted by the notation (if it is clear from the context that the space has a Euclidean structure).
To define Euclidean space, it is easiest to take as the basic concept dot product. Euclidean vector space is defined as finite-dimensional vector space above field real numbers, on whose vectors it is given real-valued function having the following three properties:
Affine space corresponding to such a vector space is called a Euclidean affine space, or simply Euclidean space .
An example of Euclidean space is a coordinate space consisting of all possible n-ok real numbers, the scalar product in which is determined by the formula
Basis and vector coordinates
Basis (Old Greekβασις, basis) - a set of such vectors V vector space, that any vector of this space can be uniquely represented in the form linear combination vectors from this set - basis vectors.
In the case when the basis is infinite, the concept of “linear combination” requires clarification. This leads to two main types of definition:
Hamel basis, whose definition considers only finite linear combinations. The Hamel basis is used mainly in abstract algebra (in particular, linear algebra).
Schauder basis, the definition of which also considers infinite linear combinations, namely, the expansion in ranks. This definition is used mainly in functional analysis, in particular for Hilbert space,
In finite-dimensional spaces, both types of basis coincide.
Vector coordinates— coefficients of the only possible linear combination basic vectors in the selected coordinate system, equal to this vector.
where are the coordinates of the vector.
Scalar product.
surgery on two vectors, the result of which is number[when considering vectors, numbers are often called scalars], independent of the coordinate system and characterizing the lengths of the factor vectors and corner between them. This operation corresponds to multiplication length vector x on projection vector y to vector x. This operation is usually considered as commutative And linear for each factor.
Scalar product two vectors is equal to the sum of the products of their corresponding coordinates:
Vector artwork
This pseudovector, perpendicular plane constructed from two factors, which is the result binary operation"vector multiplication" over vectors in three dimensions Euclidean space. The cross product does not have properties commutativity And associativity(is anticommutative) and, unlike scalar product of vectors, is a vector. Widely used in many engineering and physics applications. For example, angular momentum And Lorentz force mathematically written as a vector product. The cross product is useful for "measuring" the perpendicularity of vectors - the modulus of the cross product of two vectors is equal to the product of their moduli if they are perpendicular, and decreases to zero if the vectors are parallel or antiparallel.
Vector artwork two vectors can be calculated using determinant matrices
Mixed work
Mixed product vectors -scalar product vector on vector product vectors And:
Sometimes it is called triple scalar product vectors, apparently due to the fact that the result is scalar(more precisely - pseudoscalar).
Geometric meaning: The modulus of the mixed product is numerically equal to the volume parallelepiped, educated vectors .mixed work three vectors can be found through the determinant
Plane in space
Plane - algebraic surface first order: in Cartesian coordinate system plane can be specified equation first degree.
Some characteristic properties of the plane
Plane - surface, containing completely each direct, connecting any of it points;
The two planes are either parallel or intersect in a straight line.
A straight line is either parallel to the plane, or intersects it at one point, or is on the plane.
Two lines perpendicular to the same plane are parallel to each other.
Two planes perpendicular to the same line are parallel to each other.
Likewise segment And interval, a plane that does not include extreme points can be called an interval plane, or an open plane.
General equation (complete) of the plane
where and are constants, and at the same time they are not equal to zero; V vector form:
where is the radius vector of the point, vector 
perpendicular to the plane (normal vector). Guidescosines
vector:
Scalars can be added, multiplied, and divided just like regular numbers.
Since a vector is characterized not only by a numerical value, but also by a direction, the addition of vectors does not obey the rules for adding numbers. For example, let the lengths of the vectors a= 3 m, b= 4 m, then a + b= 3 m + 4 m = 7 m. But the length of the vector \(\vec c = \vec a + \vec b\) will not be equal to 7 m (Fig. 1).
Rice. 1.
In order to construct the vector \(\vec c = \vec a + \vec b\) (Fig. 2), special rules for adding vectors are applied.
Rice. 2. And the length of the sum vector \(\vec c = \vec a + \vec b\) is determined by the cosine theorem \(c = \sqrt(a^2+b^2-2a\cdot b\cdot \cos \alpha)\ ), where \(\alpha\,\) is the angle between the vectors \(\vec a\) and \(\vec b\).
Triangle rule
In foreign literature this method is called “tail to head”.
In order to add two vectors \(\vec a\) and \(\vec b\) (Fig. 3, a) you need to move the vector \(\vec b\) parallel to itself so that its beginning coincides with the end of the vector \(\vec a\) (Fig. 3, b). Then their sum will be the vector \(\vec c\), the beginning of which coincides with the beginning of the vector \(\vec a\), and the end with the end of the vector \(\vec b\) (Fig. 3, c).
a b c Fig. 3. The result will not change if you move the vector \(\vec a\) instead of the vector \(\vec b\) (Fig. 4), i.e. \(\vec b + \vec a = \vec a + \vec b\) ( commutative property of vectors).
a b c Fig. 4. Using the triangle rule, you can add two parallel vectors \(\vec a\) and \(\vec b\) (Fig. 6, a) and \(\vec a\) and \(\vec d\) (Fig. 7, a). The sums of these vectors \(\vec c = \vec a + \vec b\) and \(\vec f = \vec a + \vec d\) are shown in Fig. 6, b and 7, b. Moreover, the modules of the vectors \(c = a + b\) and \(f=\left|a-d\right|\).
a b Fig. 6. 
a b Fig. 7. The triangle rule can be applied when adding three or more vectors. For example, \(\vec c = \vec a_1 + \vec a_2 +\vec a_3 +\vec a_4\) (Fig. 8).
Rice. 8. Parallelogram rule
In order to add two vectors \(\vec a\) and \(\vec b\) (Fig. 9, a) you need to move them parallel to themselves so that the beginnings of the vectors \(\vec a\) and \(\ vec b\) were at one point (Fig. 9, b). Then build a parallelogram whose sides will be these vectors (Fig. 9, c). Then the sum \(\vec a+ \vec b\) will be the vector \(\vec c\), the beginning of which coincides with the common beginning of the vectors, and the end with the opposite vertex of the parallelogram (Fig. 9, d).
a b 
in d Fig. 9. Vector subtraction
In order to find the difference between two vectors \(\vec a\) and \(\vec b\) (Fig. 11), you need to find the vector \(\vec c = \vec a + \left(-\vec b \right) \) (cm.
A vector is a mathematical object that is characterized by magnitude and direction (for example, acceleration, displacement), which is cast from scalars that do not have a direction (for example, distance, energy). Scalars can be added by adding their values (for example, 5 kJ of work plus 6 kJ of work equals 11 kJ of work), but vectors are not so easy to add and subtract.
Steps
Adding and subtracting vectors with known components
- Note that vectors can be one-dimensional, two-dimensional, or three-dimensional. Thus, vectors can have an "x" component, "x" and "y" components, or "x", "y", "z" components. 3D vectors are discussed below, but the process is similar for 1D and 2D vectors.
- Suppose you are given two three-dimensional vectors - vector A and vector B. Write these vectors in vector form: A =
-
To add two vectors, add their corresponding components. In other words, add the x component of the first vector to the x component of the second vector (and so on). As a result, you will get the x, y, z components of the resulting vector.
- A+B =
- Let's add vectors A and B. A =<5, 9, -10>and B =<17, -3, -2>. A+B=<5+17, 9+-3, -10+-2>, or <22, 6, -12> .
- A+B =
-
To subtract one vector from another, you need to subtract the corresponding components. As will be shown below, subtraction can be replaced by adding one vector and the inverse vector of another. If the components of two vectors are known, subtract the corresponding components of one vector from the components of the other.
- A-B =
- Subtract vectors A and B. A =<18, 5, 3>and B =<-10, 9, -10>. A - B =<18--10, 5-9, 3--10>, or <28, -4, 13> .
Graphic addition and subtraction
-
Since vectors have magnitude and direction, they have a beginning and an end (a starting point and an ending point, the distance between which is equal to the value of the vector). When a vector is graphically displayed, it is drawn as an arrow, the tip of which is the end of the vector, and the opposite point is the beginning of the vector.
- When plotting vectors, plot all angles very accurately; otherwise you will get the wrong answer.
-
To add vectors, draw them so that the end of each previous vector is connected to the beginning of the next vector. If you're only adding two vectors, then that's all you have to do before finding the resulting vector.
- Please note that the order in which the vectors are connected is not important, that is, vector A + vector B = vector B + vector A.
-
To subtract a vector, simply add the inverse vector, that is, reverse the direction of the subtracted vector, and then connect its beginning to the end of another vector. In other words, to subtract a vector, rotate it 180 o (around the origin) and add it to another vector.
If you add or subtract how many (more than two) vectors, then connect their ends and beginnings in series. The order in which you connect the vectors does not matter. This method can be used for any number of vectors.
-
Draw a new vector, starting from the beginning of the first vector and ending with the end of the last vector (the number of vectors added is not important). You will get a resulting vector equal to the sum of all added vectors. Note that this vector is the same as the vector obtained by adding the x, y, and z components of all vectors.
- If you have drawn the lengths of the vectors and the angles between them very precisely, then you can find the value of the resulting vector simply by measuring its length. Additionally, you can measure the angle (between the resultant vector and another specified vector or horizontal/vertical lines) to find the direction of the resultant vector.
- If you have drawn the lengths of the vectors and the angles between them very accurately, then you can find the value of the resulting vector using trigonometry, namely the sine theorem or the cosine theorem. If you are adding multiple vectors (more than two), first add two vectors, then add the resulting vector and the third vector, and so on. See the next section for more information.
-
Present the resulting vector, indicating its value and direction. As noted above, if you have drawn the lengths of the vectors being added and the angles between them very precisely, then the value of the resulting vector is equal to its length, and the direction is the angle between it and the vertical or horizontal line. To the vector value, do not forget to assign the units of measurement in which the vectors to be added/subtracted are given.
- For example, if you add velocity vectors measured in m/s, then add “m/s” to the value of the resulting vector, and also indicate the angle of the resulting vector in the format “o to the horizontal line.”
Adding and subtracting vectors by finding the values of their components
-
To find the values of the vector components, you need to know the values of the vectors themselves and their direction (angle relative to a horizontal or vertical line). Consider a two-dimensional vector. Make it the hypotenuse of a right triangle, then the legs (parallel to the X and Y axes) of this triangle will be the vector components. These components can be thought of as two vectors connected, which when added together give the original vector.
- The lengths (values) of the two components (the x and y components) of the original vector can be calculated using trigonometry. If "x" is the value (modulus) of the original vector, then the vector component adjacent to the angle of the original vector is xcosθ, and the vector component opposite to the angle of the original vector is xsinθ.
- It is important to note the direction of the components. If a component is directed opposite to the direction of one of the axes, then its value will be negative, for example, if on a two-dimensional coordinate plane the component is directed to the left or down.
- For example, given a vector with a module (value) of 3 and a direction of 135 o (relative to the horizontal). Then the "x" component is equal to 3cos 135 = -2.12, and the "y" component is equal to 3sin135 = 2.12.
-
Once you have found the components of all the vectors being added, simply add their values and find the component values of the resulting vector. First, add up the values of all the horizontal components (that is, the components parallel to the X-axis). Then add up the values of all the vertical components (that is, the components parallel to the Y axis). If the value of a component is negative, it is subtracted rather than added.
- For example, let's add the vector<-2,12, 2,12>and vector<5,78, -9>. The resulting vector will be like this<-2,12 + 5,78, 2,12-9>or<3,66, -6,88>.
-
Calculate the length (value) of the resulting vector using the Pythagorean theorem: c 2 =a 2 +b 2 (since the triangle formed by the original vector and its components is rectangular). In this case, the legs are the “x” and “y” components of the resulting vector, and the hypotenuse is the resulting vector itself.
- For example, if in our example you added up the force measured in Newtons, then write the answer as follows: 7.79 N at an angle of -61.99 o (to the horizontal axis).
- Don't confuse vectors with their moduli (values).
- Vectors that have the same direction can be added or subtracted simply by adding or subtracting their values. If two oppositely directed vectors are added, their values are subtracted rather than added.
- Vectors that are represented as x i+ y j+ z k can be added or subtracted by simply adding or subtracting the corresponding coefficients. Also write the answer in the form i,j,k.
- The value of a vector in three-dimensional space can be found using the formula a 2 =b 2 +c 2 +d 2, Where a- vector value, b, c, And d- vector components.
- Column vectors can be added/subtracted by adding/subtracting the corresponding values in each row.
- A-B =
Since vectors have magnitude and direction, they can be decomposed into components based on the x, y and/or z dimensions. They are usually designated in the same way as points in a coordinate system (for example,<х,у,z>). If the components are known, then adding/subtracting vectors is as simple as adding/subtracting x, y, z coordinates.
In this article we will look at the operations that can be performed with vectors on the plane and in space. Next, we list the properties of operations on vectors and justify them using geometric constructions. We will also show the use of the properties of operations on vectors when simplifying expressions containing vectors.
To better assimilate the material, we recommend refreshing your memory of the concepts given in the article, vectors - basic definitions.
Page navigation.
The operation of adding two vectors is the triangle rule.
Let's show you how it happens addition of two vectors.
The addition of vectors occurs like this: from an arbitrary point A a vector equal to is deposited, then from a point B a vector equal to is deposited, and the vector is sum of vectors and. This method of adding two vectors is called triangle rule.
Let us illustrate the addition of non-collinear vectors on a plane according to the triangle rule.
And the drawing below shows the addition of co-directed and oppositely directed vectors.
Addition of several vectors - polygon rule.
Based on the considered operation of adding two vectors, we can add three or more vectors. In this case, the first two vectors are added, the third vector is added to the resulting result, the fourth is added to the resulting result, and so on.
The addition of several vectors is performed by the following construction. From an arbitrary point A of a plane or space a vector equal to the first term is laid off, a vector equal to the second term is laid off from its end, a vector equal to the second term is laid off from its end, and so on. Let point B be the end of the last deferred vector. The sum of all these vectors will be the vector .
Adding several vectors on a plane in this way is called polygon rule. Here is an illustration of the polygon rule.
The addition of several vectors in space is performed in exactly the same way.
The operation of multiplying a vector by a number.
Now let's figure out how it happens multiplying a vector by a number.
Multiplying a vector by the number k corresponds to vector stretching by a factor of k for k > 1 or compression by a factor of 0< k < 1 , при k = 1 вектор остается прежним (для отрицательных k еще изменяется направление на противоположное). Если произвольный вектор умножить на ноль, то получим нулевой вектор. Произведение нулевого вектора и произвольного числа есть нулевой вектор.
For example, when multiplying a vector by the number 2, we should double its length and maintain the same direction, and when multiplying a vector by minus one third, we should reduce its length by three times and change the direction to the opposite. Let us give an illustration of this case for clarity.
Properties of operations on vectors.
So, we have defined the operation of adding vectors and the operation of multiplying a vector by a number. Moreover, for any vectors and arbitrary real numbers, the following can be justified using geometric constructions: properties of operations on vectors. Some of them are obvious.
The considered properties give us the opportunity to transform vector expressions.
The commutative and associative properties of the vector addition operation allow you to add vectors in any order.
There is no operation of subtracting vectors as such, since the difference between vectors is the sum of vectors and .
Taking into account the considered properties of operations on vectors, we can perform transformations in expressions containing sums, differences of vectors and products of vectors by numbers in the same way as in numerical expressions.
Let's look at it with an example.
Consider a vector v with initial point at origin
in any x-y coordinate system and with end point at (a,b). We say that the vector is in standard position
and refer to it as the radius vector. Note that a pair of points define this vector. So we can use this to denote a vector. To emphasize that we mean a vector, and to avoid confusion, we usually write:
v = .
The coordinate a is scalar horizontal component vector, and the coordinate b is scalar vertical component vector. Under scalar we mean numerical quantity, not vector size. So this is considered as component form v. Note that a and b are NOT vectors and should not be confused with the definition of a vector component.
Now consider with A = (x 1, y 1) and C = (x 2, y 2). Let's look at how to find the radius vector equivalent to . As you can see in the picture below, the starting point A has been moved to the origin (0, 0). The coordinates of P are found by subtracting the coordinates of A from the coordinates of C. Thus, P = (x 2 - x 1, y 2 - y 1) and the radius vector is . 
It can be shown that and have the same magnitude and direction, and are therefore equivalent. Thus = = .
Component form
with A = (x 1 , y 1) and C = (x 2 , y 2) there is
= .
Example 1 Find the component form if C = (- 4, - 3) and F = (1, 5).
Solution We have
= = .
Please note that the vector is equal to the radius vector, as shown in the figure above.
Now that we know how to write a vector in component form, let's lay out some definitions.
The length of a vector v is easy to determine when the components of the vector are known. For v = , we have
|v| 2 = v 2 1 + v 2 2 Using the Pythagorean theorem
|v| = √v 2 1 + v 2 2 . 
Length , or magnitude vector v = is found as |v| = √v 2 1 + v 2 2 .
Two vectors equal or equivalent if they have the same magnitude and the same direction.
Let u = and v = . Then
= only if u 1 = v 1 and u 2 = v 2 .
Operations with vectors
To multiply a vector V by a positive number, we multiply its length by that number. Its direction remains the same. When a vector V is multiplied by 2, for example, its length doubles but its direction does not change. When a vector is multiplied by 1.6, its length increases by 60%, but its direction remains the same. To multiply a vector V by a negative real number, we multiply its length by that number and reverse the direction. For example, When a vector is multiplied by (-2), its length is doubled and its direction is reversed. Since real numbers act as scalar factors in vector multiplication, we call them scalars
and the product kv is called scalar multiples
v. 
For a real number k and a vector v = , scalar product
k and v are
kv = k. = .
Vector kv is scalar multiple
vector v.
Example 2 Let u = and w = . Find - 7w, 3u and - 1w.
Solution
- 7w = - 7. = ,
3u = 3. = ,
- 1w = - 1. = .
Now we can add two vectors using components. To add two vectors in component form, we add the corresponding components. Let u = and v = . Then
u+v= 
For example, if v = and w = , then
v + w = =
If u = and v = , then
u + v = .
Before we define vector subtraction we need to define - v. The opposite of the vector v = shown below is
- v = (- 1).v = (- 1) = 
Subtraction of vectors such as u - v involves subtracting the corresponding components. We show this by representing u - v as u + (- v). If u = and v = , then
u - v = u + (- v) = + = =
We can illustrate vector subtraction using a parallelogram, just as we did for vector addition. 
Vector subtraction
If u = and v = , then
u - v = .
It is interesting to compare the sums of two vectors with the difference of the same two vectors in one parallelogram. The vectors u + v and u - v are the diagonals of the parallelogram. 
Example 3 Do the following calculations where u = and v = .
a) u + v
b) u - 6v
c)3u + 4v
d)|5v - 2u|
Solution
a) u + v = + = = ;
b)u - 6v = - 6. = - = ;
c) 3u + 4v = 3. + 4. = + = ;
d) |5v - 2u| = |5. - 2.| = | - | = || = √(- 29) 2 + 21 2 = √1282 ≈ 35.8
Before formulating the properties of vector addition and multiplication, we must define another special vector - the zero vector. A vector whose starting point is the same as its ending point is called zero vector
, denoted O, or. Its value is 0. In vector addition:
v + O = v. + =
Operations on vectors have the same properties as operations on real numbers.
Properties of vector addition and multiplication
For all vectors u, v, and w, and for all scalars b and c:
1. u + v = v + u.
2. u + (v + w) = (u + v) + w.
3. v + O = v.
4 1.v = v; 0.v = O.
5. v + (- v) = O.
6. b(cv) = (bc)v.
7. (b + c)v = bv + cv.
8. b(u + v) = bu + bv.
Orty
A vector of magnitude or length 1 is called ort
. Vector v = is a unit vector, because
|v| = || = √(- 3/5) 2 + (4/5) 2 = √9/25 + 16/25 = √25/25 = √1 = 1.
Example 4 Find the unit vector that has the same direction as the vector w = .
Solution Let's first find the length w:
|w| = √(- 3) 2 + 5 2 = √34 . So we are looking for a vector with length 1/√34 of w and with the same direction as w. This vector is
u = w/√34 = /√34 = 34.5/√34 >.
The vector u is a unit vector because
|u| = |w/√34 | = 
= √9/34 + 25/34
= √34/34
= √1
= 1.
If v is a vector and v ≠ O, then
(1/|v|). v, or v/|v|,
There is ort
in the v direction.
Although vectors can have any direction, vectors parallel to the x and y axes are especially useful. They are defined as
i = and j = .
Any vector can be expressed as linear combination
unit vectors i and j. For example, let v = . Then
v = = + = v 1 + v 2 = v 1 i + v 2 j.
Example 5 Express the vector r = as a linear combination of i and j.
Solution
r = = 2i + (- 6)j = 2i - 6j.
Example 6 Write the vector q = - i + 7j in component form.
Solution q = - i + 7j = -1i + 7j =
Vector operations can also be performed when vectors are written as linear i and j.
Example 7 If a = 5i - 2j and b = -i + 8j, find 3a - b.
Solution
3a - b = 3(5i - 2j) - (- i + 8j) = 15i - 6j + i - 8j = 16i - 14j.
Viewing Angles
The end point P of the unit vector in standard position is the point on the unit circle defined by (cosθ, sinθ). Thus, the unit vector can be expressed in component form,
u = ,
or as a linear combination of unit vectors i and j,
u = (cosθ)i + (sinθ)j,
where the components u are functions viewing angle
θ measured counterclockwise from the x-axis to this vector. Since θ varies from 0 to 2π, point P traces the circle x 2 + y 2 = 1. This covers all possible directions of the unit vectors and then the equation u = (cosθ)i + (sinθ)j describes every possible unit vector in the plane.
Example 8 Calculate and sketch the unit vector u = (cosθ)i + (sinθ)j for θ = 2π/3. Draw a unit circle in your sketch.
Solution
u = (cos(2π/3))i + (sin(2π/3))j = (- 1/2)i + (√3 /2)j
Let v = with viewing angle θ. Using the definition of the tangent function, we can determine the viewing angle of their components v: 
Example 9 Determine the viewing angle θ of the vector w = - 4i - 3j.
Solution We know that
w = - 4i - 3j = .
Thus we have
tanθ = (- 3)/(- 4) = 3/4 and θ = tan - 1 (3/4).
Since w is in the third quadrant, we know that θ is the angle of the third quadrant. The corresponding angle is
tan - 1 (3/4) ≈ 37°, and θ ≈ 180° + 37°, or 217°.
This is useful for working with applied problems, and in later courses, to have a way to express a vector so that its magnitude and direction can be easily determined or read. Let v be a vector. Then v/|v| is a unit vector in the same direction as v. Thus we have
v/|v| = (cosθ)i + (sinθ)j
v = |v|[(cosθ)i + (sinθ)j] Multiplying by |v|
v = |v|(cosθ)i + |v|(sinθ)j.
Angles between vectors
When a vector is multiplied by a scalar, the result is a vector. When two vectors are added, the result is also a vector. So we might expect that the product of two vectors is a vector, but it is not. Scalar product two vectors is a real number or a scalar. This result is useful in finding the angle between two vectors and in determining whether two vectors are perpendicular.
Scalar product two vectors u = and v = is
u. v = u 1 .v 1 + u 2 .v 2
(Note that u 1 v 1 + u 2 v 2 is scalar, not a vector.)
Example 10 Find the dot product when
u = , v = and w = .
a)u. w
b)w. v
Solution
a)u. w = 2(- 3) + (- 5)1 = - 6 - 5 = - 11;
b)w. v = (- 3)0 + 1(4) = 0 + 4 = 4.
The dot product can be used to find the angle between two vectors. Corner between two vectors is the smallest positive angle formed by two directed segments. Thus, θ between u and v is the same angle as between v and u, and 0 ≤ θ ≤ π.
If θ is the angle between two non-zero vectors u and v, then
cosθ = (u . v)/|u||v|.
Example 11 Find the angle between u = and v = .
Solution Let's start by finding u. v, |u|, and |v|:
u. v = 3(- 4) + 7(2) = 2,
|u| = √3 2 + 7 2 = √58 , and
|v| = √(- 4) 2 + 2 2 = √20 .
Then
cosα = (u . v)/|u||v| = 2/√58 .√20
α = cos - 1 (2/√58 .√20 )
α ≈ 86.6°.
Balance of Power
When multiple forces act on the same point on an object, their vector sum must be zero in order for there to be balance. When there is a balance of forces, the object is stationary or moving in a straight line, without acceleration. The fact that the vector sum must be equal to zero output to obtain balance, and vice versa, allows us to solve many applied problems involving forces.
Example 12 Hanging block The 350-pound unit is suspended by two cables. left. At point A there are three forces acting like this: the W block pulls down, and the R and S (two cables) pull up and out. Find the load of each cable.
Solution Let's draw a diagram with the starting points of each vector at the origin. For balance, the sum of the vectors should be equal to O: 
R + S + W = O.
We can express each vector in terms of its magnitude and viewing angle:
R = |R|[(cos125°)i + (sin125°)j],
S = |S|[(cos37°)i + (sin37°)j], and
W = |W|[(cos270°)i + (sin270°)j]
= 350(cos270°)i + 350(sin270°)j
= -350j cos270° = 0; sin270° = - 1.
Substituting R, S, and W in R + S + W + O, we have
[|R|(cos125°) + |S|(cos37°)]i + [|R|(sin125°) + |S|(sin37°) - 350]j = 0i + 0j.
This gives us a system of equations:
|R|(cos125°) + |S|(cos37°) = 0,
|R|(sin125°) + |S|(sin37°) - 350 = 0.
Solving this system, we get
|R| ≈ 280 and |S| ≈ 201.
Thus, the load on the cables is 280 lbs and 201 lbs.