Vector - a value, which is characterized by its numerical value and direction. In other words, the vector - is directed segment.
The position of the vector AB in the space defined by the coordinates of the start point of the vector A and the end point of the vector B. Consider how to determine the coordinates of the middle of the vector.
To start to define the symbols and startend of the vector. If the vector is written as AB, it is the start point of the vector A and the point B - end. Conversely, for the vector BA is the beginning point of the vector B and the point A - end. Suppose we are given a vector start coordinates AB vector A = (a1, a2, a3) and the end of the vector B = (b1, b2, b3). Then the coordinates of the vector AB will be as follows: AB = (b1 - a1, b2 - a2, b3 - a3), ie the coordinates of the end of the vector must be deducted the corresponding coordinate of the beginning of the vector. The length of the vector AB (or module) is calculated as the square root of the sum of the squares of its origin: | AB | = V ((b1 - a1) ^ 2 + (b2 - a2) ^ 2 + (b3 - a3) ^ 2).
We find the coordinates of the point, which is the middlevector. We denote it by the letter O = (o1, o2, o3). Are the coordinates of the middle of the vector as well as the coordinates of the middle of the normal segment by the following formulas: o1 = (a1 + b1) / 2, o2 = (a2 + b2) / 2, o3 = (a3 + b3) / 2. We find the coordinates of AO: AO = (o1 - a1, o2 - a2, o3 - a3) = ((b1 - a1) / 2, (b2 - a2) / 2, (b3 - a3) / 2).
Consider an example. Let a vector AB vector start coordinates A = (1, 3, 5) and the end of the vector B = (3, 5, 7). Then, the coordinates of the vector can be written as AB AB = (3 - 1 5 - 3, 7 - 5) = (2, 2, 2). We find the unit vector AB: | AB | = V (4 + 4 + 4) = 2 * v3. The value of the specified vector length will help us to further verify the coordinates of the middle of the vector. Next, find the coordinates of the point O: O = ((1 + 3) / 2, (3 + 5) / 2, (5 + 7) / 2) = (2, 4, 6). Then the coordinates of AO AO is calculated as = (2 - 1, 4 - 3, 6 - 5) = (1, 1, 1).
Perform the checks. The length of the vector AO = v (1 + 1 + 1) = v3. Recall that the original length of the vector is equal to 2 * v3, ie vector half really is half the original length of the vector. Now we calculate the coordinates of the vector OB: OB = (3 - 2 5 - 4 7 - 6) = (1, 1, 1). Find the sum vectors AO and OB: AO + OB = (1 + 1, 1 + 1, 1 + 1) = (2, 2, 2) = AB. Consequently, the coordinates of the middle right of the vector was found.