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Remark 4.4. If points M 1 and M 2 belong to plane Oxy, then coordinates of
point M can be calculated according to the first two formulas (for x and y).

Example 4.2. Three vertices of the triangle are set: A = (2; 1), B = (4; −3),
C = (−1; 2). Find the length of a median, draw from point C towards side
AB (fig. 4.14).

Solution. Since median CM divides side AB into halves, then point M is the midpoint of interval AB.
So, coordinates of point M are: x = 2+4 = 3, y = 1−3 = −1. Then, the length of median CM can be
2 2
√
2
2
calculated according to the formula of the length of the interval: |CM| = (3 + 1) + (−1 − 2) = 5.
B

A . . . . . M C

Figure 4.14 – Illustration to Example 4.2

Lecture 5. Multiplication of Vectors

5.1. Dot product of vectors

− →
−→
Let’s consider two non-zero vectors a and b .

Definition 5.1. A digit, which equals to the multiple of modules of vectors a and
−→
−→
b and cosine of angle φ between these vectors, is called the dot product of vectors
a and b . It is denoted by: a · b (fig. 5.1).
−→ −→ − → − →

So, according to the definition,

−→
−→ − → −→
a · b = | a | · b · cos φ (5.1)
−→
In case at least one of vectors equals to zero, then the angle between vectors is φ b indefinite,
and the dot product is considered to be equal to zero.
Let’s consider a physical task that leads to a dot product of vectors. Suppose that point
M is moving along a straight line from point A towards point B, having done a certain way
−→
S. Suppose, point M is under the impact of constant force F , which is directed under angleα

−→
towards this point. It is known, work W can be calculated by the formula: W = F · S · cos α.
−→ − → −→
According to the definition, the last formula can be written as: W = F · S . Thus, work of

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