ov, first you need to understand such a concept as deferring a vector from a given point.
Definition 1
If point $A$ is the beginning of any vector $\overrightarrow(a)$, then the vector $\overrightarrow(a)$ is said to be delayed from point $A$ (Fig. 1).
Figure 1. $\overrightarrow(a)$ plotted from point $A$
Let us introduce the following theorem:
Theorem 1
From any point $K$ one can plot a vector $\overrightarrow(a)$ and, moreover, only one.
Proof.
Existence: There are two cases to consider here:
Vector $\overrightarrow(a)$ is zero.
In this case, it is obvious that the desired vector is the vector $\overrightarrow(KK)$.
Vector $\overrightarrow(a)$ is non-zero.
Let us denote by the point $A$ the beginning of the vector $\overrightarrow(a)$, and by the point $B$ the end of the vector $\overrightarrow(a)$. Let us draw a straight line $b$ through the point $K$ parallel to the vector $\overrightarrow(a)$. Let us plot the segments $\left|KL\right|=|AB|$ and $\left|KM\right|=|AB|$ on this line. Consider the vectors $\overrightarrow(KL)$ and $\overrightarrow(KM)$. Of these two vectors, the desired one will be the one that will be co-directed with the vector $\overrightarrow(a)$ (Fig. 2)
Figure 2. Illustration of Theorem 1
Uniqueness: uniqueness immediately follows from the construction carried out in the “existence” point.
The theorem is proven.
Subtraction of vectors. Rule one
Let us be given vectors $\overrightarrow(a)$ and $\overrightarrow(b)$.
Definition 2
The difference of two vectors $\overrightarrow(a)$ and $\overrightarrow(b)$ is a vector $\overrightarrow(c)$ which, when added to the vector $\overrightarrow(b)$, gives the vector $\overrightarrow(a)$ , that is
\[\overrightarrow(b)+\overrightarrow(c)=\overrightarrow(a)\]
Designation:$\overrightarrow(a)-\overrightarrow(b)=\overrightarrow(c)$.
Let's consider constructing the difference between two vectors using the problem.
Example 1
Let the vectors $\overrightarrow(a)$ and $\overrightarrow(b)$ be given. Construct the vector $\overrightarrow(a)-\overrightarrow(b)$.
Solution.
Let's construct an arbitrary point $O$ and plot the vectors $\overrightarrow(OA)=\overrightarrow(a)$ and $\overrightarrow(OB)=\overrightarrow(b)$ from it. By connecting point $B$ with point $A$, we obtain the vector $\overrightarrow(BA)$ (Fig. 3).
Figure 3. Difference of two vectors
Using the triangle rule for constructing the sum of two vectors, we see that
\[\overrightarrow(OB)+\overrightarrow(BA)=\overrightarrow(OA)\]
\[\overrightarrow(b)+\overrightarrow(BA)=\overrightarrow(a)\]
From Definition 2, we get that
\[\overrightarrow(a)-\overrightarrow(b)=\overrightarrow(BA)\]
Answer:$\overrightarrow(a)-\overrightarrow(b)=\overrightarrow(BA)$.
From this problem we obtain the following rule for finding the difference of two vectors. To find the difference $\overrightarrow(a)-\overrightarrow(b)$ you need to plot the vectors $\overrightarrow(OA)=\overrightarrow(a)$ and $\overrightarrow(OB)=\overrightarrow(b) from an arbitrary point $O$ )$ and connect the end of the second vector to the end of the first vector.
Subtraction of vectors. Rule two
Let us remember the following concept we need.
Definition 3
The vector $\overrightarrow(a_1)$ is called arbitrary for the vector $\overrightarrow(a)$ if these vectors are opposite in direction and have equal length.
Designation: Vector $(-\overrightarrow(a))$ is the opposite of vector $\overrightarrow(a)$.
In order to introduce the second rule for the difference of two vectors, we first need to introduce and prove the following theorem.
Theorem 2
For any two vectors $\overrightarrow(a)$ and $\overrightarrow(b)$ the following equality holds:
\[\overrightarrow(a)-\overrightarrow(b)=\overrightarrow(a)+(-\overrightarrow(b))\]
Proof.
By definition 2, we have
We add the vector $\left(-\overrightarrow(b)\right)$ to both parts, we get
Since the vectors $\overrightarrow(b)$ and $\left(-\overrightarrow(b)\right)$ are opposite, then $\overrightarrow(b)+\left(-\overrightarrow(b)\right)=\overrightarrow (0)$. We have
The theorem is proven.
From this theorem we obtain the following rule for the difference between two vectors: To find the difference $\overrightarrow(a)-\overrightarrow(b)$, we need to plot the vector $\overrightarrow(OA)=\overrightarrow(a)$ from an arbitrary point $O$, then, from the resulting point $A$, plot the vector $\overrightarrow(AB)=-\overrightarrow(b)$ and connect the beginning of the first vector with the end of the second vector.
Example of a problem on the concept of vector difference
Example 2
Let a parallelogram $ADCD$ be given whose diagonals intersect at point $O$. $\overrightarrow(AB)=\overrightarrow(a)$, $\overrightarrow(AD)=\overrightarrow(b)$ (Fig. 4). Express the following vectors through the vectors $\overrightarrow(a)$ and $\overrightarrow(b)$:
a) $\overrightarrow(DC)+\overrightarrow(CB)$
b) $\overrightarrow(BO)-\overrightarrow(OC)$
Figure 4. Parallelogram
Solution.
a) We perform the addition according to the triangle rule, we get
\[\overrightarrow(DC)+\overrightarrow(CB)=\overrightarrow(DB)\]
From the first rule for the difference of two vectors, we get
\[\overrightarrow(DB)=\overrightarrow(a)-\overrightarrow(b)\]
b) Since $\overrightarrow(OC)=\overrightarrow(AO)$, we get
\[\overrightarrow(BO)-\overrightarrow(OC)=\overrightarrow(BO)-\overrightarrow(AO)\]
By Theorem 2, we have
\[\overrightarrow(BO)-\overrightarrow(AO)=\overrightarrow(BO)+\left(-\overrightarrow(AO)\right)=\overrightarrow(BO)+\overrightarrow(OA)\]
Using the triangle rule, we finally have
\[\overrightarrow(BO)+\overrightarrow(OA)=\overrightarrow(BA)=-\overrightarrow(AB)=-\overrightarrow(a)\]
How vector addition occurs is not always clear to students. Children have no idea what is hidden behind them. You just have to remember the rules, and not think about the essence. Therefore, it is precisely the principles of addition and subtraction of vector quantities that require a lot of knowledge.
The addition of two or more vectors always results in one more. Moreover, it will always be the same, regardless of how it is found.
Most often, in a school geometry course, the addition of two vectors is considered. It can be performed according to the triangle or parallelogram rule. These drawings look different, but the result of the action is the same.
How does addition occur using the triangle rule?
It is used when the vectors are non-collinear. That is, they do not lie on the same straight line or on parallel ones.
In this case, the first vector must be plotted from some arbitrary point. From its end it is required to draw parallel and equal to the second. The result will be a vector starting from the beginning of the first and ending at the end of the second. The pattern resembles a triangle. Hence the name of the rule.
If the vectors are collinear, then this rule can also be applied. Only the drawing will be located along one line.
How is addition performed using the parallelogram rule?
Again? applies only to non-collinear vectors. The construction is carried out according to a different principle. Although the beginning is the same. We need to set aside the first vector. And from its beginning - the second. Based on them, complete the parallelogram and draw a diagonal from the beginning of both vectors. This will be the result. This is how vector addition is performed according to the parallelogram rule.
So far there have been two. But what if there are 3 or 10 of them? Use the following technique.
How and when does the polygon rule apply?
If you need to perform addition of vectors, the number of which is more than two, do not be afraid. It is enough to put them all aside sequentially and connect the beginning of the chain with its end. This vector will be the required sum.
What properties are valid for operations with vectors?
About the zero vector. Which states that when added to it, the original is obtained.
About the opposite vector. That is, about one that has the opposite direction and equal magnitude. Their sum will be zero.
On the commutativity of addition. Something that has been known since elementary school. Changing the positions of the terms does not change the result. In other words, it doesn't matter which vector to put off first. The answer will still be correct and unique.
On the associativity of addition. This law allows you to add any vectors from a triple in pairs and add a third to them. If you write this using symbols, you get the following:
first + (second + third) = second + (first + third) = third + (first + second).
What is known about vector difference?
There is no separate subtraction operation. This is due to the fact that it is essentially addition. Only the second of them is given the opposite direction. And then everything is done as if adding vectors were considered. Therefore, there is practically no talk about their difference.
In order to simplify the work with their subtraction, the triangle rule is modified. Now (when subtracting) the second vector must be set aside from the beginning of the first. The answer will be the one that connects the end point of the minuend with the same one as the subtrahend. Although you can postpone it as described earlier, simply by changing the direction of the second.
How to find the sum and difference of vectors in coordinates?
The problem gives the coordinates of the vectors and requires finding out their values for the final result. In this case, there is no need to perform constructions. That is, you can use simple formulas that describe the rule for adding vectors. They look like this:
a (x, y, z) + b (k, l, m) = c (x + k, y + l, z + m);
a (x, y, z) -b (k, l, m) = c (x-k, y-l, z-m).
It is easy to see that the coordinates simply need to be added or subtracted depending on the specific task.
First example with solution
Condition. Given a rectangle ABCD. Its sides are equal to 6 and 8 cm. The intersection point of the diagonals is designated by the letter O. It is required to calculate the difference between the vectors AO and VO.
Solution. First you need to draw these vectors. They are directed from the vertices of the rectangle to the point of intersection of the diagonals.
If you look closely at the drawing, you can see that the vectors are already combined so that the second of them is in contact with the end of the first. It's just that his direction is wrong. It should start from this point. This is if the vectors are added, but the problem involves subtraction. Stop. This action means that you need to add the oppositely directed vector. This means that VO needs to be replaced with OV. And it turns out that the two vectors have already formed a pair of sides from the triangle rule. Therefore, the result of their addition, that is, the desired difference, is the vector AB.
And it coincides with the side of the rectangle. To write down your numerical answer, you will need the following. Draw a rectangle lengthwise so that the larger side is horizontal. Start numbering the vertices from the bottom left and go counterclockwise. Then the length of vector AB will be 8 cm.
Answer. The difference between AO and VO is 8 cm.
Second example and its detailed solution
Condition. The diagonals of the rhombus ABCD are 12 and 16 cm. The point of their intersection is designated by the letter O. Calculate the length of the vector formed by the difference between the vectors AO and BO.
Solution. Let the designation of the vertices of the rhombus be the same as in the previous problem. Similar to the solution to the first example, it turns out that the required difference is equal to the vector AB. And its length is unknown. Solving the problem came down to calculating one of the sides of the rhombus.
For this purpose, you will need to consider the triangle ABO. It is rectangular because the diagonals of a rhombus intersect at an angle of 90 degrees. And its legs are equal to half the diagonals. That is, 6 and 8 cm. The side sought in the problem coincides with the hypotenuse in this triangle.
To find it you will need the Pythagorean theorem. The square of the hypotenuse will be equal to the sum of the numbers 6 2 and 8 2. After squaring, the values obtained are: 36 and 64. Their sum is 100. It follows that the hypotenuse is equal to 10 cm.
Answer. The difference between the vectors AO and VO is 10 cm.
Third example with detailed solution
Condition. Calculate the difference and sum of two vectors. Their coordinates are known: the first one has 1 and 2, the second one has 4 and 8.
Solution. To find the sum you will need to add the first and second coordinates in pairs. The result will be the numbers 5 and 10. The answer will be a vector with coordinates (5; 10).
For the difference, you need to subtract the coordinates. After performing this action, the numbers -3 and -6 will be obtained. They will be the coordinates of the desired vector.
Answer. The sum of the vectors is (5; 10), their difference is (-3; -6).
Fourth example
Condition. The length of the vector AB is 6 cm, BC is 8 cm. The second is laid off from the end of the first at an angle of 90 degrees. Calculate: a) the difference between the modules of the vectors VA and BC and the module of the difference between VA and BC; b) the sum of the same modules and the module of the sum.
Solution: a) The lengths of the vectors are already given in the problem. Therefore, calculating their difference is not difficult. 6 - 8 = -2. The situation with the difference module is somewhat more complicated. First you need to find out which vector will be the result of the subtraction. For this purpose, the vector BA, which is directed in the opposite direction AB, should be set aside. Then draw the vector BC from its end, directing it in the direction opposite to the original one. The result of subtraction is the vector CA. Its modulus can be calculated using the Pythagorean theorem. Simple calculations lead to a value of 10 cm.
b) The sum of the moduli of the vectors is equal to 14 cm. To find the second answer, some transformation will be required. Vector BA is oppositely directed to that given - AB. Both vectors are directed from the same point. In this situation, you can use the parallelogram rule. The result of the addition will be a diagonal, and not just a parallelogram, but a rectangle. Its diagonals are equal, which means that the modulus of the sum is the same as in the previous paragraph.
Answer: a) -2 and 10 cm; b) 14 and 10 cm.
In mathematics and physics, students and schoolchildren often come across problems involving vector quantities and performing various operations on them. What is the difference between vector quantities and the scalar quantities we are used to, the only characteristic of which is their numerical value? The fact is that they have direction.
The use of vector quantities is explained most clearly in physics. The simplest examples are forces (frictional force, elastic force, weight), speed and acceleration, since in addition to numerical values they also have a direction of action. For comparison, let's give example of scalar quantities: This can be the distance between two points or the mass of a body. Why is it necessary to perform operations on vector quantities such as addition or subtraction? This is necessary so that it is possible to determine the result of the action of a vector system consisting of 2 or more elements.
Definitions of vector mathematics
Let us introduce the main definitions used when performing linear operations.
- A vector is a directed segment (having a beginning point and an end point).
- Length (modulus) is the length of the directed segment.
- Collinear are two vectors that are either parallel to the same line or simultaneously lie on it.
- Oppositely directed vectors are called collinear and at the same time directed in different directions. If their direction coincides, then they are codirectional.
- Vectors are equal when they are co-directional and identical in magnitude.
- The sum of two vectors a And b is such a vector c, the beginning of which coincides with the beginning of the first, and the end with the end of the second, provided that b starts at the same point where it ends a.
- Vector difference a And b name the amount a And ( - b ), Where ( - b ) - oppositely directed to the vector b. Also, the definition of the difference between two vectors can be given as follows: the difference c pairs of vectors a And b they call this c, which when added to the subtrahend b forms a minuend a.
Analytical method
The analytical method involves obtaining the coordinates of the difference using a formula without plotting. It is possible to perform calculations for flat (two-dimensional), volumetric (three-dimensional) or n-dimensional space.
For two-dimensional space and vector quantities a {a₁;a₂) And b {b₁;b₂} the calculations will look like this: c {c₁; c₂} = {a₁ – b₁; a₂ – b₂}.
In the case of adding a third coordinate, the calculation will be carried out similarly, and for a {a₁;a₂; a₃) And b {b₁;b₂; b₃) the coordinates of the difference will also be obtained by pairwise subtraction: c {c₁; c₂; c₃} = {a₁ – b₁; a₂ – b₂; a₃ – b₃}.
Calculating the difference graphically
In order to construct the difference graphically, you should use the triangle rule. To do this, you must perform the following sequence of actions:
- Using the given coordinates, construct vectors for which you need to find the difference.
- Combine their ends (that is, construct two directed segments equal to the given ones, which will end at the same point).
- Connect the beginnings of both directed segments and indicate the direction; the resultant will begin at the same point where the vector being minuend began and end at the point where the subtrahend begins.
The result of the subtraction operation is shown in the figure below.
There is also a method for constructing the difference, which differs slightly from the previous one. Its essence lies in the application of the vector difference theorem, which is formulated as follows: in order to find the difference of a pair of directed segments, it is enough to find the sum of the first of them with a segment oppositely directed to the second. The construction algorithm will look like this:
- Construct the initial directed segments.
- The one that is subtracted must be reflected, that is, construct a segment oppositely directed and equal to it; then combine its beginning with the minuend.
- Construct a sum: connect the beginning of the first segment with the end of the second.
The result of this decision is shown in the figure:
Problem solving
To consolidate the skill, we will analyze several tasks in which you need to calculate the difference analytically or graphically.
Problem 1. There are 4 points given on the plane: A (1; -3), B (0; 4), C (5; 8), D (-3; 2). Determine the coordinates of the vector q = AB - CD, and also calculate its length.
Solution. First you need to find the coordinates AB And CD. To do this, subtract the coordinates of the initial points from the coordinates of the end points. For AB the beginning is A(1; -3), and the end – B(0; 4). Let's calculate the coordinates of the directed segment:
AB {0 - 1; 4 - (- 3)} = {- 1; 7}
A similar calculation is performed for CD:
CD {- 3 - 5; 2 - 8} = {- 8; - 6}
Now, knowing the coordinates, you can find the difference between the vectors. The formula for the analytical solution of plane problems was considered earlier: for c = a- b coordinates have the form ( c₁; c₂} = {a₁ – b₁; a₂ – b₂). For a specific case, you can write:
q = {- 1 - 8; 7 - (- 6)} = { - 9; - 1}
To find the length q, let's use the formula | q| = √(q₁² + q₂²) = √((- 9)² + (- 1)²) = √(81 + 1) = √82 ≈ 9.06.
Problem 2. The figure shows the vectors m, n and p.
It is necessary to construct differences for them: p- n; m- n; m-n- p. Find out which of them has the smallest modulus.
Solution. The problem requires three constructions. Let's look at each part of the task in more detail.
Part 1. In order to depict p- n, Let's use the triangle rule. To do this, using parallel translation, we connect the segments so that their end point coincides. Now let's connect the starting points and determine the direction. In our case, the difference vector begins in the same place as the subtrahend n.
Part 2. Let's depict m - n. Now to solve we will use the vector difference theorem. To do this, construct a vector opposite n, and then find its sum with m. The resulting result will look like this:
Part 3. In order to find the difference m - n - p, you should split the expression into two actions. Since vector algebra has laws similar to the laws of arithmetic, the following options are possible:
- m - (n + p): in this case, the sum is first plotted n+p, which is then subtracted from m;
- (m - n) - p: here you first need to find m - n, and then subtract from this difference p;
- (m - p) - n: the first action is determined m - p, after which you need to subtract from the result obtained n.
Since in the previous part of the problem we already found the difference m - n, we just have to subtract from it p. Let's construct the difference between two given vectors using the difference theorem. The answer is shown in the image below (red indicates the intermediate result, and green indicates the final result).
It remains to determine which of the segments has the smallest modulus. Let us remember that the concepts of length and modulus in vector mathematics are identical. Let's visually estimate the lengths p- n, m-n And m-n-p. Obviously, the shortest answer and the one with the smallest modulus is the answer in the last part of the problem, namely m-n-p.
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.
X and y called a vector z such that z+y=x.
Option 1. The starting points of all vectors coincide with the origin of coordinates.
Let's construct the difference of vectors and 
.
To plot the vector difference z=x-y, you need to add the vector x with the opposite to y vector y". Opposite vector y" it's easy to build:
Vector y" is opposite to the vector y, because y+y"= 0, where 0 is a zero vector of the appropriate size. Next, vector addition is performed x And y":
From expression (1) it is clear that to construct the difference between vectors, it is enough to calculate the differences in the corresponding coordinates of the vectors x And y.
Rice. 1
In the picture Fig. 1 in two-dimensional space the difference of vectors is represented x=(10.3) and y=(2,4).
Let's calculate z=x-y=(10-3,3-4)=(7,-1). Let us compare the obtained result with the geometric interpretation. Indeed, after constructing the vector y" and parallel movement of the starting point of the vector y" to the end point of the vector x, we get the vector y"", and after adding vectors x And y"", we get the vector z.
Option 2. The starting points of the vectors are arbitrary.
Rice. 2
In the picture Fig. 2 in two-dimensional space the difference of vectors is represented x=AB And y=CD, Where A(1,0), B(11,3), C(1,2), D(3.6). To calculate the vector z=x-y, constructed opposite to the vector y vector y":
|
Next you need to add the vectors x And y". Vector y" moves parallel so that the point C" coincided with the point B. To do this, the differences in the coordinates of the points are calculated B And WITH.
