Instruction

1

To start with the notations of the beginning and end

**of the vector**. If the vector is written as AB, the point A is the beginning**of the vector**, and point B – end. Conversely, for a**vector**BA, the point B is the beginning**of the vector**, and point A the end. Let us set the vector AB with coordinates of beginning**of vector**A = (a1, a2, a3) and the end**of the vector**B = (b1, b2, b3). Then the coordinates**of vector**AB are as follows: AB = (b1 – a1, b2 – a2, b3 – a3), i.e., the coordinates of the end**vector**it is necessary to subtract the corresponding coordinate of the beginning**of the vector**. Length**of vector**AB (or module) is calculated as the square root of the sum of the squares of its coordinates: |AB| = √((b1 – a1)^2 + (b2 – a2)^2 + (b3 – a3)^2).2

Find the coordinates of the midpoint

**of the vector**. Let's denote it with the letter O = (o1, o2, o3). Are the coordinates of the means**vector**as well as the coordinates of the mid-cut normal, according to the following formulas: o1 = (a1 + b1)/2, o2 = (a2 + b2)/2 , o3 = (a3 + b3)/2. Find the coordinates**of the vector**AO: AO = (o1 – a1, o2 – a2, A3 – a3) = ((b1 – a1)/2, (b2 – a2)/2, (b3 – a3)/2).3

Let's consider an example. Let vector AB given coordinates of beginning

**of vector**A = (1, 3, 5) and the end**of the vector**B = (3, 5, 7). Then the coordinates**of the vector**AB can be written as AB = (3 – 1, 5 – 3, 7 – 5) = (2, 2, 2). Find the module**of the vector**AB: |AB| = √(4 + 4 + 4) = 2 * √3. The length of the specified**vector**will help us to further validate the correctness of 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 the vector**calculated as AO AO = (2 – 1, 4 – 3, 6 – 5) = (1, 1, 1).4

Run the test. Length

**of the vector**AO = √(1 + 1 + 1) = √3. Recall that the length of the original**vector**is equal to 2 * √3, i.e. half**of the vector**is indeed equal to half the length of the original**vector**. Now let's calculate the coordinates**of the vector**OB: OB = (3 – 2, 5 – 4, 7 – 6) = (1, 1, 1). Find the sum of vectors AO and OB: AO + OB = (1 + 1, 1 + 1, 1 + 1) = (2, 2, 2) = AB. Therefore, the coordinates of the middle of the**vector**were found true.Useful advice

Following the calculation of the coordinates of the middle of the vector, be sure to perform at least the simplest test is to count the length of the vector and compare it with the length of this vector.