The rules by which the addition of vectors
How is the addition of vectors, it is not always clear to students. Children do not represent what is hidden behind them. You just have to memorize the rules, and not ponder the essence. Therefore, it is about the principles of addition and subtraction that a lot of knowledge is required.
As a result of the addition of two or more vectors, one is always obtained. Moreover, it will always be the same, regardless of the reception of its location.
Most often in the school course of geometry is considered the addition of two vectors. It can be performed by the rule of a triangle or parallelogram. These figures look different, but the result of the action is the same.
How is the addition of the rule of the triangle?
It is applied when vectors are noncollinear. That is, do not lie on one line or on parallel lines.
In this case, from some arbitrary point you need to postpone the first vector. From its end it is required to conduct a parallel and equal to the second.The result will be a vector originating from the beginning of the first and ending at the end of the second. Figure 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 the addition according to the parallelogram rule?
Yet again? applies only to non-collinear vectors. The construction is carried out according to another principle. Although the beginning is the same. It is necessary to postpone 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. She will be the result. This is the way to add vectors according to the parallelogram rule.
So far, there have been two. And what if there are 3 or 10? Use the following trick.
How and when does the polygon rule apply?
If you want to perform the addition of vectors, the number of which is more than two, you should not be afraid. It is enough to postpone them all sequentially and connect the beginning of the chain with its end. This vector will be the desired sum.
What properties are valid for actions with vectors?
About zero vector.Which asserts that when added with it, the original is obtained.
On the opposite vector.That is, one that has the opposite direction and is equal in magnitude to the value. Their sum will be equal to zero.
On commutativity of addition.What is known since elementary school. Changing the places of the items does not change the result. In other words, no matter which vector to put off first. The answer will still be true and unique.
On the associativity of addition.This law allows you to add in pairs any vectors from a triple and add a third one to them. If you write it with the help of signs, you get the following:
first + (second + third) = second + (first + third) = third + (first + second).
What is known about the difference of vectors?
A separate subtraction operation does not exist. This is due to the fact that it is, in fact, an addition. Only the second of them is given the opposite direction. And then everything is done as if the addition of vectors were considered. Therefore, they practically do not speak about their differences.
In order to simplify the work with their subtraction, the triangle rule has been modified. Now (when subtracting) the second vector must be postponed from the beginning of the first. The answer will be the one that connects the end point of the deductible with it.Although it is possible to postpone 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 you want to know their values for the final. In this construction is not necessary to perform. That is, you can use simple formulas that describe the rule of addition of vectors. They look like this:
a (x, y, z) + in (k, l, m) = c (x + k, y + l, z + m);
a (x, y, z) -c (k, l, m) = c (xk, y-l, z-m).
It is easy to notice that the coordinates you just need to add or subtract, depending on the specific task.
The first example with the solution
Condition. Given a rectangle AVSD. Its sides are 6 and 8 cm. The intersection point of the diagonals is indicated by the letter O. It is required to calculate the difference of the vectors AO and VO.
Decision. First you need to draw these vectors. They are directed from the vertices of the rectangle to the intersection point of the diagonals.
If you look closely at the drawing, you can see that the vectors are already aligned so that the second of them is in contact with the end of the first. That's just his direction is wrong. It must start from this point. This is if the vectors are added up, and in the problem - subtraction. Stop. This action means that you need to add an oppositely directed vector. This means that VO needs to be replaced with OB.And it turns out that two vectors have already formed a pair of sides from the rule of a triangle. Therefore, the result of their addition, that is, the desired difference, is the vector AB.
And it coincides with the side of the rectangle. In order to record a numerical answer, the following will be required. Draw a rectangle along so that the big side goes horizontally. The numbering of the vertices start from the bottom left and go counterclockwise. Then the length of the vector AB will be equal to 8 cm.
Answer. The difference between AO and VO is 8 cm.
The second example and its detailed solution
Condition. The rhombus AVSD diagonal is equal to 12 and 16 cm. The point of their intersection is denoted by the letter O. Calculate the length of the vector formed by the difference of the vectors AO and VO.
Decision. Let the designation of the vertices of a rhombus be the same as in the previous problem. Similarly to the solution of the first example, it turns out that the desired difference is equal to the vector AB. And its length is unknown. The solution of the problem was reduced to calculating one of the sides of the rhombus.
For this purpose, you need to consider the triangle ABO. It is rectangular, because the diagonal of the rhombus intersects at an angle of 90 degrees. And his 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, we need the Pythagorean theorem. The square of the hypotenuse will be equal to the sum of the numbers 62and 82. After squaring, the values are 36 and 64. Their sum is 100. From this it follows that the hypotenuse is 10 cm.
Answer. The difference between the vectors of AO and HE is 10 cm.
The third example with a detailed solution
Condition. Calculate the difference and the sum of two vectors. Their coordinates are known: in the first - 1 and 2, in the second - 4 and 8.
Decision. To find the amount you will need to add in pairs the first and second coordinates. 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 perform the subtraction of coordinates. After performing this action, you get the numbers -3 and -6. They will be the coordinates of the desired vector.
Answer. The sum of vectors is (5; 10), their difference is (-3; -6).
Condition. The length of the vector AB is 6 cm, BC - 8 cm. The second one is plotted from the end of the first at an angle of 90 degrees. Calculate: a) the difference of the modules of the vectors BA and BC and the module of the difference BA and BC; b) the sum of the same modules and the modulus of the sum.
Solution: a) The lengths of the vectors are already given in the problem. Therefore, to calculate their difference is not difficult.6 - 8 = -2. The situation with the difference module is somewhat more complicated. First you need to know which vector will be the result of subtraction. For this purpose, you should postpone the vector BA, which is directed in the opposite direction AB. Then from its end to hold the vector of the sun, directing it in the direction opposite to the original. The result of the subtraction is the vector CA. Its module can be calculated by the Pythagorean theorem. Simple calculations lead to a value of 10 cm.
b) The sum of the modules of the vectors is 14 cm. To search for the second answer, some conversion is required. Vector BA is oppositely directed to that given by - AB. Both vectors are directed from one 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.