The area of the parallelogram constructed on vectors is calculated as the work of lengths of these vectors on a sine of the angle between them. If only coordinates of vectors are known, then it is necessary to apply coordinate methods including for definition of a corner between vectors to calculation.
It is required to you
- - concept of a vector;
- - properties of vectors;
- - Cartesian coordinates;
- - trigonometrical functions.
Instruction
1. In case lengths of vectors and a corner between them are known, then to find the area of the parallelogram constructed on vectors find the work of their modules (lengths of vectors), on a sine of the angle of S= between them │ a│ • │ b│ • sin(α).
2. If vectors are set in the Cartesian system of coordinates, then to find the area of the parallelogram constructed on them do the following actions:
3. Find coordinates of vectors if they are not given at once, having taken away from the corresponding coordinates of the ends of vectors, coordinate from the beginnings. For example, if coordinates of the initial point of a vector (1;-3;2), and final (2;-4;-5), coordinates of a vector will be (2-1;-4+3;-5-2) (1;-1;-7). Let coordinates of a vector and (x1; y1; z1), b vector (x2; y2; z2).
4. Find lengths of each of vectors. Square each of coordinates of vectors, find their sum x1²+y1²+z1². Take a root from the turned-out result square. For the second vector do the same procedure. Thus, it will turn out ai │ b │.
5. Find a scalar product of vectors. For this purpose multiply their corresponding coordinates and put the works │a b │ = x1 • x2 + y1 • y2 + z1 • z2.
6. Define a cosine of the angle between them for what the scalar product of vectors which turned out in item 3 divide into the work of lengths of vectors which were calculated in item 2 (Cos(α) = by │a b │ / (│ a│ • │ b │)).
7. The sine of the received corner will be equal to a root square of the difference of number 1, and a square of a cosine of the same corner calculated in item 4 (1-Cos² (α)).
8. Calculate the area of the parallelogram constructed on vectors having found the work of their lengths calculated in item 2, and increase result by the number which turned out after calculations in item 5.
9. In case coordinates of vectors are set on the plane, when calculating the coordinate of z is just rejected. This calculation is numerical expression of the vector work of two vectors.