The vector as the directed piece, depends not only on an absolute value (module) which is equal to its length. One more important characteristic – the direction of a vector. It can be defined both by coordinates, and a corner between a vector and an axis of coordinates. Calculation of a vector is also made when finding the sum and the difference of vectors.
It is required to you
- - definition of a vector;
- - properties of vectors;
- - calculator;
- - Bradis's table or personal computer.
Instruction
1. To calculate vector, it is possible knowing its coordinates. For this purpose determine coordinates of the beginning and the end of a vector. Let they will be equal (x1; y1) and (x2; y2). To make calculation of a vector, find its coordinates. For this purpose from coordinates of the end of a vector take away coordinates of its beginning. They will be equal (x2 x1; y2-y1). Accept x = x2 x1; y = y2-y1, then coordinates of a vector will be equal (x; y).
2. Determine vector length. It can be done simply, having measured it by a ruler. But if vector coordinates are known, calculate length. For this purpose find the sum of squares of coordinates of a vector and take from the turned-out number a root square. Then length of a vector will be equal to d= √ (x²+y²).
3. After that find the direction of a vector. For this purpose define a corner α between it and an axis OH. The tangent of this corner is equal to the vector y coordinate relation to coordinate x (tg α = to y/x). To find a corner, use function of an arctangent, Bradis's table or the personal computer in the calculator. Knowing length of a vector and its direction concerning an axis, it is possible to find position in space of any vector.
4. Example: coordinates of the beginning of a vector are equal (-3;5), and coordinates of the end (1;7). Find coordinates of a vector (1-(-3); 7-5) (4;2). Then its length will be d= √ (4²+2²)= √ 20≈4,47 linear units. The tangent of angle between a vector and an axis OH will make tg α=2/4=0.5. The arctangent of this corner is in round figures equal 26.6º.
5. Find a vector which represents the sum of two vectors which coordinates are known. For this purpose put the corresponding coordinates of vectors which develop. If coordinates of vectors which develop are equal according to (x1; y1) and (x2; y2), their sum will be equal to a vector with coordinates ((x1+x2; y1+y2)). If it is necessary to find the difference of two vectors, then you find the sum, previously having increased coordinates of a vector which is subtracted on-1.
6. If lengths of vectors of d1 and d2, and a corner between them α are known, find their sum, using the theorem of cosines. For this purpose find the sum of squares of lengths of vectors, and subtract the doubled work of these lengths increased by a cosine of the angle between them from the turned-out number. Take a root from the turned-out number square. It will also be length of the vector which is the sum of two data of vectors (d= √ (d1²+d2²-d1∙d2∙Cos(α)).