As the addition of vectors occurs, it is not always clear to students. Children do not represent what is behind them. You just have to memorize the rules, and do not ponder the essence. Therefore, it is about the principles of addition and subtraction of vector quantities that require a lot of knowledge.
As a result of the addition of two or more vectors, there is always one more. And he will always be the same regardless of the reception of his location.
Most often in the school course of geometry, the addition of two vectors is considered. It can be executed by the rule of a triangle or a parallelogram. These pictures look different, but the result from the action is one.
How does addition work according to the rule of the triangle?
It is applied when the vectors are noncollinear. That is, do not lie on one line or parallel.
In this case, it is necessary to postpone the first vector from some arbitrary point. From its end it is required to draw a parallel and equal to the second. The result is a vector originating 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 done by the parallelogram rule?
Yet again? It is used only for noncollinear vectors. The construction is done in a different way. Although the beginning is the same. It is necessary to postpone the first vector. And from its beginning - the second. On their basis, complete the parallelogram and draw a diagonal from the beginning of both vectors. It will be the result. This is how the vectors are added by the parallelogram rule.
So far there have been two. And what if there are 3 or 10? Use the following method.
How and when does the polygon rule apply?
If you want to add vectors, the number of which is more than two, you should not be frightened. It is sufficient to sequentially postpone them all and connect the beginning of the chain with its end. This vector u is the required sum.
What properties are valid for actions with vectors?
On the zero vector. Which states that when adding to it, the original is obtained.
On the opposite vector. That is, about one that has the opposite direction and a value equal in absolute value. Their sum will be zero.
On the commutativity of addition. What is known since primary school. Changing the places of the summands does not lead to a change in the result. In other words, it does not matter which vector to postpone 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 the triple and add a third to them. If you record this with signs, you get the following:
first + (second + third) = second + (first + third) = third + (first + second).
What is known about the difference of vectors?
There is no separate subtraction operation. This is due to the fact that it, in fact, is an addition. Only the second of them is given the opposite direction. And then everything is executed as if the addition of vectors were considered. Therefore, almost no difference is said about their difference.
In order to simplify the work with their subtraction, the rule of the triangle is modified. Now (when subtracting) the second vector must be postponed from the beginning of the first one. The answer is the one that connects the end point of the subtrahend with it. Although it is possible to postpone as described earlier, simply changing the direction of the second.
How to find the sum and difference of vectors in coordinates?
The vector coordinates are given in the problem and you need to know their values for the final one. At the same time, you do not need to perform any construction. That is, you can use simple formulas that describe the rule for adding vectors. They look like this:
a (x, y, z) + in (k, l, m) = c (x + k, y + 1, z + m);
a (x, y, z) -c (k, l, m) = c (x-k, y-1, z-m).
It is easy to see that the coordinates simply need to be added or subtracted depending on the specific task.
The first example with the solution
Condition. A rectangle of ABCD is given. Its sides are 6 and 8 cm. The intersection point of the diagonals is denoted by the letter O. It is required to calculate the difference between the vectors AO and BO.
Decision. First you need to depict 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 aligned so that the second one touches the end of the first one. That's just his direction is wrong. He must start from this point. This is if the vectors add up, and in the task - subtraction. Stop. This action means that you need to add an oppositely directed vector. Hence, VO must be replaced by OB. And it turns out that the two vectors have already formed a pair of sides from the rule of the 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 write down a numerical answer, the following is required. Draw a rectangle along so that the larger side runs horizontally. Numbering of the vertices begin with the bottom left and go counter-clockwise. Then the length of the vector AB will be 8 cm.
Answer. The difference between AO and BO is 8 cm.
The second example and its detailed solution
Condition. The diamonds of the ABCD diamond are 12 and 16 cm. The point of their intersection is indicated by the letter O. Calculate the length of the vector formed by the difference between the vectors AO and BO.
Decision. Let the notation of the vertices of the rhombus be the same as in the previous problem. Similar to the solution of the first example, 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, we need to consider the triangle ABO. It is rectangular, because the diagonals of the rhombus intersect 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 task 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 6 2 and 8 2. After squaring, the values are 36 and 64. Their sum is 100. It follows that the hypotenuse is 10 cm.
Answer. The difference between the vectors AO and BO is 10 cm.
The third example with a detailed solution
Condition. Calculate the difference and sum of two vectors. Their coordinates are known: the first - 1 and 2, the second - 4 and 8.
Decision. To find the sum, it is required to combine the first and second coordinates in pairs. The result will be the numbers 5 and 10. The answer is a vector with coordinates (5; 10).
For the difference, you need to subtract the coordinates. After doing this, the numbers -3 and -6 are obtained. They are the coordinates of the required vector.
Answer. The sum of the vectors is (5; 10), their difference is (-3; -6).
Condition. The length of the vector AB is 6 cm, BC - 8 cm. The second is laid from the end of the first at an angle of 90 degrees. Calculate: a) the difference between the moduli of the vectors BA and BC and the modulus of the difference between BA and BC; b) the sum of these modules and the modulus of the sum.
Solution: a) The lengths of the vectors are already given in the problem. Therefore, it is not difficult to calculate their difference. 6 - 8 = -2. The situation with the difference modulus is somewhat more complicated. First you need to find out which vector will be the result of subtraction. For this purpose, it is necessary to postpone the VA vector, which is directed in the opposite direction of AB. Then from its end, draw the vector VS, pointing it to the side opposite to the original one. The result of subtraction is the CA vector. Its module can be calculated by the Pythagorean theorem. Simple calculations lead to a value of 10 cm.
b) The sum of the moduli of the vectors is found to be 14 cm. To search for the second answer, some transformation is required. The vector VA is opposite to the one given - AB. Both vectors are directed from one point. In this situation, you can use the parallelogram rule. The result of addition will be a diagonal, not just a parallelogram, but a rectangle. Its diagonals are equal, so the modulus of the sum is the same as in the previous paragraph.
The answer is: a) -2 and 10 cm; b) 14 and 10 cm.